Time dilation: why do the observers see each other as the slow one but then one of them is older or younger than the other? I'm in trouble with time dilation:
Suppose that there are two people, A and B, on the Earth. They are twins and each has a clock. They are therefore in the same reference frame). B travels in a spaceship and is orbiting the Earth, and as B's speed has increased there's some time dilation. So it's supposed that if A "looks" at B, B's clock will run slower than A's, and vice versa. And if one day B decides come back to the Earth, he will be younger than his twin and the time shown on his clock would be different (earlier than A's clock) because of time dilation.
My first question is, why does each person see the other's clock as the slow one? (In fact, I found some information that may have solved it but I'm not sure if it's the right answer, and if it is, I don't understand it. -> It's because everyone thinks of himself as the reference frame. In other words, that for B, he is the stationary one and the one who moves is A, so he sees A's clock run slower as B is "stopped" and the same for A. B is who is moving and A the stationary.)
The other question I have is that if it's true that each one sees the other as the slow one there's something that I missed out. We all agree that B (after going to space and coming back to the Earth) is younger than his brother. So, why doesn't B see A moving and aging very fast if A sees his brother and his things going very slow? Let's imagine they were looking at each other all the time, and if both of them see the other going slowly, and then they meet and "discover" that B is the young one... how is that possible?
[I think that in one episode of Cosmos by Carl Sagan (though I may be mistaken), Sagan said that a neutrino "borned" in the Big Bang could have seen the creation of the Universe until today in just a few seconds due to its high speed, here start my doubts: or I misunderstood something or there's contradictory information]
(I don't know anything about physics, only what is taught at high school, as I'm a teenager, so it'd be better to answer with no kind of calculation as I'd not understand it.)
 A: There are lots of questions about the twin paradox on this site, so it's probably not worth going over that material again.
What is worth saying is that where people tend to get confused is by misunderstanding what an inertial frame is and how different inertial frames can be compared. We should simplify matters a bit and put twin B on a spaceship because orbital motion is a bit more complicated. The only time A and B can directly compare anything with each other is the moment that they pass i.e. the moment that they are in the same place. If A and B stay in their inertial frames they will never meet again and indeed will get further and further apart as time passes. The only way the twins will ever meet again is if they change inertial frames i.e. if one of them accelerates.
In SR acceleration is absolute. By this I mean that velocity is relative i.e. you cannot tell whether A or B is the one moving, but it is always possible to tell which of the two is accelerating. The acceleration always introduces an assymetry so it's no surprise that when they meet again A and B will find their clocks differ.
You can treat acceleration in special relativity. See for example my answer to How do I adjust the kinematic equations to avoid reaching speeds faster than light? where I give (some of) the equations for understanding the motion of an accelerating rocket. If you do the calculation you will find that B sees A clock running fast while B is accelerating between inertial frames. See my answer to Why isn't the symmetric twin paradox a paradox? for more on this.
The question of what Carl Sagan's neutrino sees is quite a subtle one. Suppose some particle interaction shortly after the Big Bang and 13.7 billion years away produced a neutrino and that neutrino has just passed you. For the neutrino only a few seconds has passed since the Big Bang. However when the neutrino passes you it sees you at your current age, 13.7 billion years, so what's going on? The answer is that in the neutrino's frame and your frame the Big Bang happened at different times. So the neutrino can see the 13.7 billion age of the Big Bang pass in a few seconds, but not because it sees the universe's time running fast. It sees each successive bit of the universe as older because in each bit of the universe it passes through the Big Bang gets further and further back in time.
A: First, let me say that I really like your statement, "[...] so it'd be better to answer with no kind of calculation as I'd not understand it." I think I might start saying that myself when I ask questions about relativity.
Like you, I rarely understand any explanation of relativity when it is based on equations. In the rare cases when I can understand what is being expressed in the equations, I am unable to tell whether the math is valid. I like reading explanations that use words and pictures, and I like explaining things in words and pictures. I have mastered the basics of relativity, so I'll try to give you an explanation of time dilation and the twins "paradox" that you can understand. It's not so much a paradox as a counterintuitive fact of physics.
What you are asking about, essentially, is the "twins paradox" of relativity, which is a thought experiment used by teachers to teach relativity, so it's a very good question. On the other hand you have made the question more complicated than necessary by bringing orbiting into it. So I will explain only the classic twins paradox, and ignore the complicating stuff that you have added to it in your question. In so doing, I will give an easy to understand foundation for all your future thinking about special and general relativity.
Understanding Time Dilation.
Note to the reader: If you already understand exactly what a frame of reference is, what time dilation is, and what time desynchronization is, feel free to skip to the section called "Looking at the Twins Paradox in Detail".
There are a number of misconceptions about the twins paradox. The little-known truth is that it is purely a question about special relativity, and there's no need to bring general relativity into it. This is because general relativity is relevant only when gravitational forces are important parts of the question. All the rest, including accelerated frames of reference, are handled with special relativity. The truth is that acceleration by itself does not cause time dilation, few people know this.
In order for the traveling twin to turn get back to the earth, he has to have a period of acceleration, and that since he's going from close to light speed away from earth to close to light speed towards the earth, that acceleration must be very great, or last for a very long time. Many conclude that time dilation during the acceleration explains the asymmetry and thus resolves the "paradox". Not so. It's easy to show why. Imagine if the acceleration is enormously great, almost infinite, and lasts only for a second. No matter how great the time dilation during that acceleration it could only induce an age difference of one second at most. So time dilation during acceleration can't be the explanation.
In fact, acceleration does not affect the rate at which a clock runs. Moving clocks run slow, yes. On the other hand, accelerating a clock is not enough to make it run at a different rate. If a clock is both moving and accelerated, it will run slow, but not by different about compared to a clock moving with a constant velocity.
Acceleration means rate of change of velocity with respect to time. It's therefore possible to have zero velocity, and therefore zero time dilation, while having a huge acceleration. In fact, we will see that this occurs during the traveling twin's journey.
A constant velocity means a constant speed and a constant direction. Changing direction with a constant speed is a case of acceleration, because it is a case of changing velocity. It makes sense, since to change the direction of motion of an object takes a force. A familiar case is constant speed circular motion, which is caused by centripetal force.
A huge misconception that used to be very common is that moving clocks run slow because something is interfering with their function. That's why the twin "paradox" was created: having a person get older faster than their twin makes it clear that time is really passing at a different rate, in every sense of the word, "time". It's not an illusion, and it's not due to a clock experiencing some sort of friction or something.
The fact that moving rulers are short is not due to warping of the material the ruler is made of, but due to length or "space" itself changing due to motion. When the speed of light was being measured in experiments before 1905, it was noticed that it seemed that the measuring apparatus was getting shorter depending on its own speed, and they came up with a formula for calculating the amount (which I'll spare you) and name for it, which I'll tell you, the "Lorentz contraction". The scientists looked at what was known about materials and atoms to figure out how motion could could cause the Lorentz contraction. They thought the material was shrinking. Einstein was the first to realize that it was space (or length) itself that was contracting. As Einstein said, "Moving clocks run slow, moving measuring rods are short". But the clocks and rods are normal, and functioning correctly. Rather, it is that time and length (or "space") are different in a moving frame of reference. But what is a "frame of reference"?
Frames of reference are fundamental to relativity, and if you misunderstand this, you will misunderstand a lot else. Fortunately, it is easy to understand, which makes it a bit weird that so many physics graduates do not.
Here's how to understand what a frame of reference is. From now on, I'll call it a "frame" for short. Imagine a rigid framework of metal bars that are so strong and stiff that they don't bend or stretch or compress at all. They are rigidly joined to each other, and can't move relative to each other. This framework has attached it a multitude of tiny clocks that all show the same time as each other. There are clocks all over it. Any point inside the framework has a clock there.
A frame is a mathematical abstraction,  while this lattice of rods and clocks is its physical equivalent. Just a few things are bit weird about it. First, it is infinite. It fills the entire universe. Second, it can be moved about, and is weightless, infinitely light. For example, it can be attached to a rocket and if the rocket is moving then the attached lattice is moving with the same velocity. This is what is meant by the "frame of the rocket". So this framework is infinitely strong and light. Thirdly, frames of reference can through each other. Thus the frame of the earth (thought of as at rest) and the frame of the rocket is moving relative to it (without the metal bars colliding). The frame of the rocket is also covered in clocks, all showing the same time as each other. So you have two infinite lattices of rods and clocks moving relative to each other. Since all motion is relative, if the rocket is moving with a constant velocity, the twin on the rocket can say he is the one at rest, and the earth moving (relative to him). To put it another way, to the twin on the rocket, the rocket's frame is at rest, and it is the earth's frame that is moving (please imagine a nonrotating earth with a constant velocity).
When explaining the clocks and rods lattice to beginners a lot of unnecessary and, in my view, counterproductive fuss is made by teachers and others about how to synchronize the clocks of a frame with each other. If you aren't convinced, you might find this answer of mine useful, which is an answer to a question about what is the meaning of clocks and rods in relativity and which focuses on what synchronization of clocks in a clocks and rods lattice means and how it can be done in a way that is straightforward: https://physics.stackexchange.com/a/718723/295887
In a nutshell, the clocks are synchronized in the ordinary sense of the word, as in "synchronize watches", as they say in the movies.
The clocks of a frame all show the same time as each other, although never exactly the same as those of another frame. Thus all the clocks on the rocket and anywhere in its infinite frame show the same time. If you were on the rocket with the traveling twin, and you looked out the window with a powerful telescope at a physical lattice of clocks and rods attached to the rocket, you'd see a stationary lattice of clocks and rods, and the clocks nearby would tick over first, and then those further off, and then those further off still, and so on. What you see is not what is really happening. Light has a finite speed, and so it takes time for the light to get to your eye. The light takes longer to get to you from the clocks further away, so you see them tick over later. This is consistent with the fact that they are synchronized. If you saw them all ticking over at the same time, no matter regardless of distance, that would tell you that the clocks were not synchronized. This is another rarely understood truth.
More generally, the idea that relativistic effects are caused by signal delay is one of the biggest misconceptions about relativity that there is. There are entire books explaining relativity for the layman as caused by signal delay. Utter baloney, and fortunately such books are becoming less common. I repeat, relativistic effects are not caused by signal delay. A lot of interesting optical effects are caused by it, though. For example, a rocket will look like it is flying through space diagonally, due having been rotated because of moving near to the speed of light. But it's just a trick of the light. The light from the far side of the rocket arrives later than the light from the near side, making is look rotated. It isn't actually rotated. Again, this illusory rotation is frequently presented as a relativistic effect, as a real rotation, by numerous people who should know better. If only people would master what a frame is and how to use one to make valid measurements this signal delay nonsense wouldn't arise. So how do you make valid measurements with a rods and clocks and lattice?
It's simple, like everything about a clocks and rods lattice. You must prevent signal delay from messing up you measurements. That's why you have clocks everywhere. It allows you to measure the time at any point in the lattice without having to wonder is happening to the light as it moves across space from the clock to your eye. You are right at the clock. If you were blind, you could read the time off the clock by using your hands and the sense of touch. So light and all other signals are kept negligible, or you can even make all measurments by touch and thus eliminate light altogether, and with light, signal delay and all those pesky tricks of the light that are affected by moving near to the speed of light (which we have to do to get strong relativistic effects). So that's it. You measure the position by looking at the numbers painted on the rods, just like using a ruler, and you measure the time of any event (such as the passing of the front of a rocket) using a clock attached to the lattice at the point (or near as darn it to it) where the event takes place. It's a case of "what" (the point of the nose cone of a rocket), "where" (at a specific labelled location in the lattice, perhaps recorded as three numbers representing northness, eastness, and altitude), and "when" (as indicated by a clock attached to the lattice that is right at where the nose cone was when its position in (or relative to) the lattice was noted. The "when" part is the one we need to be extra careful about, and that is the whole point of the rods and clocks lattice idea.
By the way, did you ever wonder what spacetime coordinates are? Well, you just found out. Three numbers specify the location in space in a frame, and one number specifies the time. In other words, three numbers specify "where" and one specifies "when".
Notice the lattice isn't used to measure the rocket's speed, direction, or acceleration. Those can all be deduced from a set of measurements of what is where at what time. That's how the lattice works. Think of it as having a technician at each and every point in the lattice, so there are an infinite number of technicians, one for each clock. When an event happens, exactly one technician notes the position (his own) and the time (on the exactly one clock he is in charge of). He then makes a copy of the date (what, where, and when) seals it in an envelope and mails it to a bookkeeper who can be anywhere, and doesn't even need to be attached to that lattice. The bookkeeper, after receiving a bunch of envelopes from one lattice, opens them all, and plots the points on some sort of graph, and works out what happened. The bookkeeper can thus make statements that are true in that  (earth's) frame about the velocity, acceleration, size, and shape of the rocket. Likewise the same bookkeeper could do the same thing with a bunch of envelopes from the rocket's frame (maybe the envelopes are a different color), and deduce what happened where and when in the rocket's frame. This system allows us to talk about how event A happens before event B in one frame, while in another, B happens first (the technical term is "relativity of simultaneity"). The difference in the order of events (not the apparent order, the *actual order), depends on different sets of clocks being used to get each result.
Have you heard of an inertial frame? That just means a frame that is moving with a constant velocity, and is not rotating of course. Inertial frames are very important because if your lattice is rotating you will get very complicated results. If your lattices are not rotating, and all are moving with constant velocities (not all with the one velocity, but just all constant) then everything is much simpler, and so there is this nice phrase, "inertial frame", that is used a lot in relativity.
If a rocket is moving through space with a constant velocity (there's no friction, and its engines all all off), it's frame is an inertial frame. Likewise if earth is thought of as not rotating and not going around the sun, we can say that a frame of reference attached to the earth is an inertial frame.
Time dilation is a moving clock running slow because time itself has been affected, for example because the clock is moving close to the speed of light. Time dilation is part of the explanation for what happens with the twins, but, and this is crucial, it's only part of the story. There is another effect that is just a important, but compared time dilation, which a lot of people have heard of, very few people know about it. It's called "time desynchronization". This is when time itself gets skewed by moving close to speed of light. Like time dilation, it's not an effect due to a change in the functioning of the clocks, or even of the set of clocks. Within a frame the clocks are still in synch. It's only within another frame that they are no longer in synch. It's essentially the same thing as relativity of simultaneity. A set of clocks at rest that are synchronized at rest, are not synchronized in moving frame.
Looking at the Twins Paradox in Detail: A Detailed Blow by Blow Account of the Behavior of Each Twin's Frame and It's Clocks.
Let's look at the twin paradox now. In what follows, I will talk only of what actually happens in each frame, which means what a bookkeeper would work out had happened by opening a bunch of envelopes and analyzing their contents. I will not say anything about what anyone would actually see with with their eyes, because that is not what is actually happening. There are too many optical illusions when you travel at close to the speed of light.
There's a frame reference attached to a nonrotating, nonorbiting earth, and there's a different frame attached to the rocket. One twin remains on earth while the other pilots the rocket to a star ten light years away, say, and then flies straight back. the twin on earth has aged 20 years and earth's clocks all say that 20 years have passed. The traveling twin has aged by just a day, and the rocket's clocks all say that just 24 hours have passed. Lets look at this in more detail.
At first the clocks on earth and the rocket say 0 (zero). That is to say, all the clocks in the imaginary lattice of clocks and rods attached to the earth read zero. Likewise all the clocks in the frame attached to the rocket.
Suddenly the rocket starts accelerating, with the clocks in both frames saying zero, and to make it simple we can say that in the earth's frame it takes the rocket a moment to reach a speed that is a mere smidgen less than the speed of light, and then the rocket shuts off its engines. So the rocket hasn't gone far, but its velocity is very different all of a sudden, but now it is moving with a constant velocity, and it has attached to it an inertial frame. Before it started accelerating, it had an inertial frame (actually it shared a frame with earth because it had the same velocity as earth). While accelerating, it had an accelerated frame, which is to say not an inertial frame. Now the rocket has a new inertial frame.
In the rocket's frame the clocks are all a smidgen over zero, and that means everywhere in the universe including at the destination star. That's right, the rocket's frame has a time at the star. We can also say that the star is at rest in the earth's frame, and that it therefore shares a frame with the earth. I should have said that at the outset. Within the earth frame the star's clocks (which is the same thing as saying the earth frame clocks) read a smidgen over zero, a second or so, same as the clocks on earth, within the earth frame, because they are still synchronized. Within the earth frame the rocket frame clocks will be running about a millionth as fast as normal.
Within the rocket frame, moving at just under the speed of light with respect to the earth frame, the earth frame clocks (including the star's clocks) will be running about a millionth of the normal rate. This is because of time dilation.
What about time desynchronization? Well, in the rocket frame, the earth's clock (not part of the rocket frame) will be displaying zero, but the earth frame clock at the star, ie the star's clocks will be displaying ten years. Give or take a few seconds. This is time desynchronization. Because the earth clocks and the star clocks are moving at almost the speed to light with respect to the rocket, they are ten years out of synch. It's ten years because the gap between them in the earth frame is ten light years.
Why did I say "in the earth frame"? Because in the rocket frame the earth and star, besides being flattened to disks, are also only a few light hours apart. This is the Lorentz contraction. The earth to star length has been contracted because in the rocket frame it is a moving measuring rod ten light years long at rest, but now very far from at rest. And remember, their clocks are effectively frozen, because they are moving at one millionth of the normal rate.
Then, in the earth frame, the rocket takes a smidgen over ten years to travel ten light years to the star, and thus the earth frame clocks at earth and at the star display "ten years" when the rocket arrives at just under the speed of light. In the earth frame the rocket's clocks are showing a few hours. This is the effect of time dilation. The rocket is as flat as piece of paper due to being Lorentz contracted. It's clocks tick glacially, adding to a few hours ticked off during a decade of travel.
In the rocket frame, the rocket travels the few light hours that separate the flattened earth and the flattened star, which at a just under the speed of light takes a few hours. As the rocket passes the earth frame clocks laid out in a line from earth to the star, the traveling twin reads off the times, ranging from a few seconds next to earth, to five years at the half way mark, to ten years at the star, taking a few hours to pass their imperceptible glacial ticking, flattened-to-paper forms. The rocket is normal in shape and size and in every other way. The rocket arrives at the star traveling at just under the speed of light, with the rocket's clocks, in the rocket frame, showing a few hours. In the rocket frame, the earth's clocks are still at a few seconds, and the star's clocks are displaying just over ten years, and have not changed what they display by more than a few seconds during the rocket's journey.
In the rocket frame, the rocket turns on its engines and after several seconds has stopped. It is now at rest with respect with the star and the earth. The star clock is running at a normal rate, as is the earth's. The star clock still reads ten years. The earth's clock now reads ten years. The engines are still running, but the huge acceleration is not causing any time dilation. All the clocks, in both frames of reference are moving at the same rate. Notice too, that there is no additional desynchronization beyond the preexisting difference between a few hours and ten years.
In the earth frame the rocket takes about a second to stop. Its clocks run normally for a moment. The clocks on rocket display a few hours. The huge ongoing acceleration has no effect on the rate of running of any clocks. The rocket's length is suddenly normal. The acceleration continues and within a second or so, the rocket is traveling at just under the speed of light towards the earth. The earth and star clocks carry on running at a normal rate, but the rocket, now paper thin again, has clocks frozen at the same value plus a second or two that they displayed when it arrived.
In the rocket frame, as soon as cruising speed has been attained, the star and earth clocks are frozen, and paper thin. The star clock reads ten years plus a few hours as it did on arrival. The earth's clock reads 20 years, and is frozen.
In the rocket frame, the rocket makes exactly the same kind of trip as on the outward leg. A few ours pass, frozen paper thin clocks pass, displaying 11 years, 12 years, up to 19 years, and finally, on the earth clock, 20 years plus a few hours (times two). The rockets clocks displays two helpings of a few hours on arrival at earth. The clocks at the star still show ten years plus one helping of a few hours.
In the earth frame, the rocket, still traveling at just under the speed of light, arrives, having taken twenty years plus a few hours  to make the return trip to the star. It takes one second to stop. As it stops it goes from a round disk to a long cylinder. It's clocks go from frozen to normal. The twin goes from being a painted statue to a human being moving in a normal way. He opens the hatch and get out of the rocket, and gives his twin a hug. His twin is, of course, twenty year older than him. Old enough to be his father. Twenty years of history have passed. The culture is significantly different now, as is the technology.
By the way, time desynchronization also applies to the rocket. If the the rocket at rest is a light second long (300,000 km long), then when Lorentz contracted down to being a few kilometers long, the clocks on the back of the rocket will be one full second ahead of the clocks on the front, just kilometers away. This is the dramatic effect of time desynchronization.
With a longer rocket, let's call it a spaceship perhaps 100,000 light years long at rest, with enough speed, it could become one millimeter long, and the back would 100,000 years ahead of the the front. But on the spaceship itself, everything is normal, because you are moving at the same velocity as the rocket, and its frame. If a spaceship 100,000 light years long contracted to one millimeter suddenly stopped at the earth, what would happen? In the earth frame, would any part of the spaceship break the cosmic speed limit? No, because the rocket would stretch itself out to it's full length as it stopped bit by bit. It would look like a short piece of chalk that drew at just under the speed of light a line 100,000 light years long on the vacuum of space. But the line would be made of stationary spaceship, while the chalk was made of stationary spaceship. This is because the back of the spaceship stops first in the earth frame, due to time desynchronization. Then the bit a little bit further forward. In the spaceship frame, the whole spaceship stopped at the same time, and it was the earth, the solar system and the Milky Way Galaxy that returned to normal length (also looking like pieces of chalk drawing at the speed of light on the vacuum), and quit being time desynchronized, and quit moving at a snail's pace less than the speed of light.
Why does Only One Twin Experience Reduced Aging?
I hope you can see how the acceleration of the rocket, and it's consequent reversal of velocity at the star caused the traveling twin to age by only a few hours while the earth twin aged normally. The situation is not symmetrical because only the rocket twin is accelerated at any time. Only the rocket twin sees the earth frame clocks desynchronized one way for a while and then the opposite way for a while. The twin changes from one frame to another at the star. The earth does not, and the nontravelling twin is on earth the whole time and so does remains attached to the earth frame. Okay, but so what? I mean, how does it work? That I do not yet know.
A Simpler Version of the Twins Paradox, that (Kind Of) is free of acceleration.
Instead of the the traveling twin accelerating at least once (at the star, when the twin changes direction and returns to earth, you can instead imagine two rockets, one flying past earth a twenty years before the other and in the opposite direction in the earth frame. The twin on the earth has the time on his clock copied to the clock (the clock on the rocket is synchronized with the clock that the earth bound twin has) on the first passing rocket, and ten years later, the first rocket, at the star, meets the second rocket coming the other way and time on the first rockets clock is compied to the clock on the second rocket. Ten year later, the second rocket's time is copied from it's clock to a clock on the ground on earth, and is compared with the earth twin's clock. The time is much less. If the two rockets are moving at just under the speed of light, the other earth clock will show only a few days have passed, while the earthbound twin will have aged twenty years. This shows that it is nothing to do with acceleration, only having the clock, or just the information, be in one frame for a while, and then in another.
Why Oh Why does this Weird Stuff Happen at All? Why isn't Physics Newtonian?
I've described what actually happens, and which parts are due to time dilation and which parts are due to time desycnchronization. Why does any of this happen. Well, it seems that it's just the way the universe works. No one knows why it's like this. No one knows why the speed of light is the speed it is, 300,000 kilometers per second, and not greater or less.
A lot of extremely smart and clued in people who are also at a totally different level from me (and you, Esther) when it comes to math, favor an explanation that I think is quite hard to understand, and that I haven't been able to understand completely as of now. It's called the geometric explanation of time dilation, and it involves Minkowski diagrams and the latter are way over my head. But the idea is that the traveling twins world line (path through four dimensional spacetime) is different from that of the twin who remains on earth, and there is no symmetry to the situation. Anyway, you can search on Physics to find out more about Minkowski diagrams, but they are hard to visualize, and the math is really hard. Good luck and beware wrong information, as relativity seems to be plastered in wrong or misleading information. Also beware of terms being used in unfamiliar ways. For example "mass" and "energy" have been used with new meanings since about 1970 by professional relativists, or by a subset of them, at least. See Why is there a controversy on whether mass increases with speed?
