Is time dilation an illusion? It is said that we can verify time dilation by flying a very accurate clock on a fast jet or spaceship and prove that it registers less time than the clocks on earth. However, the clocks on earth would be moving relative to the clock on the spaceship, and since time always dilates and never goes faster regardless of the direction of relative motion,  the clocks on earth should register less time than the clock on the spaceship.
Is this true? Whenever there is a fast-moving object such as a rocket do all clocks on earth really become slow?
If the rocket with the clock landed after moving at relativistic speed, would its clock and the earth's clock again show the same time since during its travel both appeared slow to each other?
Or is all this just an illusion, ie. the clocks just appear to be slow to each other but in actually run at normal speed, and neither is behind when the rocket actually lands?
 A: You're referring to what is commonly known as the twin paradox. The Wikipedia page provides several different ways of analysing the situation, but one way to look at it is this - 
When the clock on the spaceship leaves earth, it'll experience an acceleration (even if it's a really small acceleration for a long time, or a huge acceleration for a small time) to reach relativistic speeds, and will experience an acceleration again when it has to turn around to return to earth.
Special relativity only claims that inertial frames of reference are equivalent. Since one clock experiences acceleration, and hence is in a non-inertial frame momentarily, the two situations aren't equivalent.
If you view the time dilation due to acceleration as a gravitational acceleration (principle of equivalence) and then do the calculations, the results obtained for both the clock on the earth as well as the clock on the spaceship agree. Again, the wikipedia link contains in detail this argument, as well as other arguments.
A: In your reasoning, you're tacitly making an assumption of a universal time, or you're at least having doubts owing to this assumption. Your central issue seems to be that two relatively moving inertial observers would each see each other's clocks moving slower to one another. This only leads to a logical contradiction if one assumes that there is one, universal time measurement, which would be shown to be strictly less than itself by your argument. Your implied argument is perfectly sound; it's the assumption of universal time that is wrong. But time is relative: each inertial observer has their own observed time, and simulteneity is relative.
So what of the case where the observers meet up again at a common spacetime point, so they have different elapsed times since they synchronized watches at the former time when they were together? Well, they have different paths through spacetime, and their clocks measure the lengths of those paths. Inertial observers fare geodesics through spacetime, which means their path is of maximal length compared to neighboring paths (this is a slightly unwonted aspect of Lorentzian geometry: in Euclidean / Riemannian geometry we deal with geodesics that are minimum length paths) - so the stay-at-home, always inertial twin in the twin "paradox" ages most. This difference between measured pathlengths joining the same two spacetime events is overwhelmingly experimentally verified, as several of the other answers show. Look up the Hafele Keating experiment, or the Rossi-Hall experiment cited in John Rennie's Answer. You might also find some insight into a more experimentally-oriented conception of time, as I discuss here.
A: Time dilation is real and is measured every day. For example the lifetime of a muon produced in the lab at low velocity is 2.2$\mu$s. However the lifetime of muons generated by high energy cosmic rays in the atmosphere is around 11$\mu$s. Their lifetime is extended by their high speed.
Calculating the time dilation of a plane flying around the Earth is complex because you have to take into account the gravitational time dilation as well as the plane's speed. However it has been done, and indeed experiment shows that if you put an atomic clock in a plane you do indeed measure the predicted time dilation.
A: If the rocket with the clock landed after moving at relativistic speed, would its clock and the earth's clock again show the same time since during its travel both appeared slow to each other?

For the rocket to depart from Earth and later return is a scenario in which a loop is closed. 
A loop-closing scenario comes out symmetrical only when both participants have traveled the same spatial distance from departure to rejoining.
In an Earth-and-a-spacecraft-on-a-relativistic-journey scenario the spacecraft travels a much longer spatial distance than the Earth.
According to special relativity for the clock that has traveled a longer spatial distance less proper time will have elapsed, as seen when comparing clock times on rejoining.
The comparison on rejoining is a direct comparison, so it's clear and unambiguous.
It's actually unhelpful to try an visualise what will be observed during the journey; because of transmission delays those raw observations are not a good perspective on what is happening. 


It may be a surprise for you that difference in spatial distance traveled matters in special relativity. You may figure 'in space any individual spacecraft cannot count how many miles it has traveled'. And that is the case: any individual spacecraft cannot count its own mileage. More forcefully, no such individual mileage exists. However, special relativity does imply that you can always evaluate difference in spatial distance traveled. That difference in spatial distance traveled must be thought of as something that is relative between the two participants.


The relativity of special relativity is not a sweeping any motion is relative to something else. There is room for structure and buildup, giving rise to non-symmetrical scenario's, such as the one you ask about.
A: No, time dilation is real. The universe does not follow the theory we assumed in the past where objects follow the Newtonian laws and light travels at a fixed speed in one frame of reference. If it did, an object going away from us would appear to have a slower system when it doesn't really.
It has been confirmed by observation that at each point in space-time, the universe locally follows special relativity. Special relativity predicts that time dilation is real. According to special relativity, there exists a frame of reference that each system dilates in time by a factor of $\frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ and contracts in length by a factor of $\frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$. According to this answer, if you're travelling at constant velocity, there's no way to tell that you're moving in that frame of reference. You can consistently assume from your observations that you are not moving and what you deduce from the assumption that you are not moving is said to have actually happened in your frame of reference. You can sometimes make observations that wouldn't be possible at all if light travelled at a fixed speed and matter followed Newton's laws.
A: "time dilation is a true physical phenomenon" As A body accelerates closer to the speed of light, all of its extremities and intrinsic energies expend velocity, "for example" if a star where traveling near the speed of light then the orbiting moon or planet would have to slow down in order to keep up with the star, or in this case, hands on a clock slowing not to exceed light speed for the rate it is travelling.
A: Time dilation is based on clock time.
Clock time is the time we measure with when doing experiments.
It is also the time according to our eyes.
In clock time, the speed of light is constant.
Most of Physics study is done using clock time.
Things age, and happen more slowly when the local rate of Clock Time is slower that the rate of Clock Time it is being compared with.
STR and GTR are theories based on on Clock Time
Clock Time theories tell us what we will see and what is happening according to our clocks.
There is an underlying time that natural activity is synchronised to - real time. One that is constant everywhere, and in which energy and momentum are independent and the speed of light is variable. It's rate is something like the rate of Earth's rotation, but without the fluctuations. A theory based on this interpretation of time can complement GR quite nicely.
Time Dilation is real insofar as it tells us the rate at which things happen in one reference frame relative to another. However, it is based on clock time, which is virtual. 
Real time doesn't, and never will vary. Things can still happen at different rates in different reference frames measured against real time. The rate of time itself is constant. 
Everything we see is some form of optical illusion.
UPDATE I will support this with an example
Jack and Jill live on the top and bottom floors of a skyscraper, respectively
When Earth rotates through one revolution, the rotation starts and stops at exactly the same time for Jack and Jill
If Jack, Jill, the skyscraper and Earth were immortal, a trillion revolutions of Earth would start and stop at exactly the same time for all four of them.
They would experience every revolution, every radian and every microarcsecond together. They would all rotate together in real time.
Jack would age round 20 minutes more than Jill and his clock would be advanced by just over twenty minutes compared with Jill's clock (based on experimental Gravitational Time Dilation data and assuming a 150m skyscraper)
This example highlights the following points:


*

*Clock Time is not the same as Real Time

*Things age more quickly at higher altitudes in Real Time

*Clock Time is variable

*Real Time is constant

*The Speed of Light is variable in Real Time

*The Speed of Light is constant in Clock Time

*Time Dilation applies to Clock Time not to Real Time

A: Gravitational or else called acceleration time dilation, has actual effects on clocks since it slows down matter vibrations and thus time evolution of matter and therefore tick rates on clocks. The larger the acceleration the slower time passes (i.e. not time itself but how time is measured). A clock coming out of acceleration into an inertial frame of reference would be found relative to lack in time. So acceleration would have caused a permanent effect on the clock (lost time) independent of the frame of reference. Gravitational or else called acceleration time dilation is an actual physical effect and phenomenon that slows down molecular vibrations on matter. This above, includes also biological clocks and the aging process of living organisms. A man on Jupiter would age much slower than a man on Earth, assuming he/she is immune to the hostile environment of Jupiter :)
Kinematic time dilation or else named constant velocity time dilation is a different phenomenon. It is an apparent time dilation effect but with real consequences on how we measure  time and our everyday life like the GPS satellites which loose in average 7μs in time each day compared to clocks on the surface of the Earth due this SR kinematic effect. At the same time they gain time about 45μs due the gravitational time dilation effect [GPS and Relativity][1] .
These two opposite effects must be accounted for 45-7= +38μs per day,  a total amount of time dilation the clocks on the satellites are going relative ahead in time (faster) compared to the clocks on the Earth surface and compensation must be applied so all clocks are in synchronization.
For the question you may rise that the velocity is not constant in the case of a GPS satellite orbiting the Earth so that it should not be considered as a SR kinematic constant velocity time dilation since velocity vector is changing all the time direction and therefore this is an acceleration and not constant velocity?
However, relativistically when the two motions are compared and not each as isolated cases, this is not true since relative to the Earth's spin which is also circular motion effectively it is the same situation as two objects moving with different constant velocities in two parallel linear straight paths in space. The GPS satellites orbit the Earth  (not Geostationary or Geosynchronous orbit) about twice every day (2 orbits in a 24h period at about 14,000 Km/h speed).
Last but not least, I must clarify what I mean with the term above "apparent" time dilation effect and phenomenon?
By this I mean that this kinematic SR time dilation effect is observed and can be measured only during flight and its due the finite speed of light c and the relative different positions in space each time of the two objects in relative motion. Due the finite and not infinite speed of light, time delays are introduced on each of the two observers residing on the two objects trying each to measure the clock reading information of the other so that at the time one observer receives the clock readout of the other due to the propagation delay of this information, an added amount of time has already passed on the clock of the observer at the time it receives this information. Therefore the observer perceives this as if his/her clock is running faster than that of the opposite observer. The effect is larger the more larger the relative speed is. This is however an apparent effect observed only during flight and has no permanent effect on matter (tick rate of clocks). Therefore, assuming that their relative constant velocity motion would come to an abrupt stop and totally neglecting the acceleration time dilation effects (i.e. gravitational time dilation effects due this abrupt stop in relative motion) then the two observers meeting back both in the same inertial frame of reference and comparing their clocks' time readouts would see no difference on the time recorded on their clocks. Their clocks will show exactly the same time!
In summary for the case of the kinematic SR time dilation, it is not that the matter of the two objects is time dilated but rather the measurement information traveling from one observer to the other that is time dilated during flight.
[1]: http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html#:~:text=Each%20satellite%20in%20the%20GPS,in%20geosynchronous%20or%20geostationary%20orbits).
