Why is acceleration directed inward when an object rotates in a circle? Somebody (in a video about physics) said that acceleration goes in if you would rotate a ball on a rope around yourself.
The other man (ex Navy SEAL, on YouTube too) said that obviously it goes out, because if you release the ball, it's going to fly in outward direction.
Then somebody said that the second man doesn't know physics; acceleration goes in.
 A: As usual, a picture is worth 1,000 words.

The object is the large dot. It rotates around the circle counterclockwise. You can see it at two different times. The arrows represent the velocity of the object, the direction indicating the direction it is moving. The acceleration is, in effect, the change between the two velocities at those two times - and in general, incorporates both the change to the direction, as well as the speed.
Note the direction of the arrows. Which way does the second arrow (counterclockwise from the first) tilt, compared to the first? Toward, or away from, the center? Thus in what direction is the tendency to accelerate?
(Note: don't let the different positions of the arrows fool you. That's part of the trick with vectors - they live in their own little "world", so to speak, and always come out of the same point therein, but that "world" is "pasted" onto the object as it moves.)
The velocity has to tilt inward, because that way it stays near the central point. As it moves forward in any direction away from the circle rim, it also needs to move a little bit inward on the next "step", so to speak, to compensate for that.
And in terms of forces, what he misses is that if you are at the circle's center and holding it by a rope, then you are providing the acceleration through the force you are applying via the rope. Which way do you have to pull to keep the object going in the circular path? Away from you, or toward you?
A: If you want an object to rotate around a point you need to change its velocity, because if you don't, the object will continue to go straight with its current velocity. The change you need for the object to stay in a circle is not a change in the magnitude of the velocity, but a change in the direction. You want the direction of the velocity to change constantly in direction of the middle point where you want your object to rotate around, in order to make the object curve towards that point instead of going straight. This change in velocity is your (centripetal) acceleration, WHICH POINTS TO THE MIDDLE (this acceleration is caused by the rope). If you were to stop accelerating towards the middle (rope breaking) there would be no change in the objects velocity and it would fly straight wherever its current velocity is pointing to.
BUT if you consider the non-inertial system (which corresponds to imagining being stuck to the rope or the object and thus seeing everything around you moving instead of you moving yourself), you can calculate that there is a force acting outwards, a so called "fictitious force". This force's acceleration is called centrifugal acceleration and corresponds exactly to the centripetal acceleration.
If you haven't heard of fictitious forces and inertial systems, ignore the second paragraph.
For better visualisation google the following in images: "centripetal force and centrifugal force".
A: 
because if you release the ball, it's going to fly in outward direction

The other man is thinking from a different frame of reference, and they're disagreeing on terminology.
When you release the ball, it travels in a straight line. This is easily shown by looking at the hammer throwing discipline, which is pretty much the perfect practical experiment to our theoretical discussion.
But the other man says "outward". That is because he is thinking that the circular path (when holding the ball) is the normal path, and the straight path (when releasing the ball) is outside the circle.
But that is not an objective frame. In fact, it's the other way around.
All objects that are not under specific forces travel in a straight line. In order to have an object travel differently, you must apply a force to it. So let's think back to our ball throwing example, but let's start from a straight line situation.
We want to make the ball curve left (and end up in a circular path). So which way do we push on the ball? Left. And because we want the path to be circular, we supply a constant left pressure on the ball (where "left" rotates as the ball rotates).
If you draw this on a diagram, you will see that this "left force" points towards the center.

The black path shows the trajectory of the ball. The red arrows are the direction the ball is traveling in. The blue arrows show you the force that you have to apply in order to makes the ball go round, i.e. "rotating" the red arrow.
The blue arrows point inward. In a better drawn diagram, they'd be pointing to the center of the circle. This means that it is an inward force.
Intuitively, we could learn this by participating in the hammer throw competition. Think about this: when the hammer thrower is spinning around, does he feel like he's performing a pulling or pushing motion?
Pulling. Because the hammer keeps trying to move in a straight line (which eventually gets further away from the thrower). To prevent that from happening, the hammer thrower pulls on the hammer, therefore applying inward force to the hammer.

As an aside, to resolve the "different frame of reference" conflict here:
The inward motion is call the centripetal force. The alleged outward motion is call centrifugal force. You'll find many opinions online that claim centrifugal force doesn't exist. And they're mostly right (though I disagree that we therefore should not talk about it at all).
Centrifugal force is actually the desire for the object to move in a straight line (which is not a force, it is the absence of force). But if you think that the "normal" trajectory is the circular one (like the Navy SEAL in your question does), then this straight line appears to be a deviation from the "normal" trajectory.
Which leads the Navy SEAL to conclude that there must be a force causing this deviation.
But he's got it the wrong way around. The circular path was the deviation, and it was kept alive because of an inward force constantly deviating the normal trajectory. When that inward force stopped, the trajectory stopped being deviated, and therefore took the "normal" path again, i.e. moving in a straight line.
Centrifugal force is a perceived force. It's not real. But because the object wants to move in a straight line and fights going in a circle, the supplier of the inward force feels as if the object is trying to "pull away" from him, which is why he perceives it as a force. But it isn't.
A: As a rule of thumb: when somebody states that something is obvious you should really doubt everything he says. Especially if he is an ex navy seal :)
Think about the ball moving in circle: Newton's first law of dynamics states that if an object is left alone, meaning: the object is not subjected to forces, it would keep moving with the same velocity. Remember that velocity is a vector, so this statement means that the object left alone would keep also the same direction of motion.
But in the case of a ball moving in circle of course its direction of motion changes with time, this must imply that the ball is subjected to a force (remember that a force $\vec{F}$ creates an acceleration $\vec{a}$ according to the second law of dynamics: $\vec{F}=m\vec{a})$. Ok, but the force pulls inward or outward? (That is analogous to asking: the acceleration is directed inward or outward?) Well think again about the velocity of the ball: as time passes the velocity curves inward, this must mean that the acceleration is directed inward.
But why then if you let the ball free it moves outward? The answer is that it doesn't really move outward, it simply begins moving in a straight line again since you are no longer applying force to it, as the first principle of dynamics states. Everything is consistent. Of course moving in a straight line in this context means moving away from the previous location of the rotational motion, so an observer has the impression of the ball moving away from the center, when the ball is as stated simply continuing his motion with the velocity it had at the time of release.
A: You can't push rope.
Intuitively, rope is only useful under tension and not compression - you can pull an object with a rope, but not push it. It should be obvious that when you swing a ball on a rope, you are pulling on the rope. You can't use just a rope to accelerate an object away from you (i.e. push something), you can only use it to accelerate an object toward you (i.e. pull something).
From this very simple fact, we can surmise that when swinging a ball on a rope, the ball is accelerating toward the center, since it is impossible for the rope to impart a force on the ball in any other direction. To suggest that the ball is accelerating outward when it's released would mean that the person provides a "push" when letting go, and that the rope is capable of transmitting such a push, both of which are false - even if the person swinging the ball does "push" when they let go, there is simply no way for a rope to transmit that push to the ball.
A: Assume that there are only two nearby things in the universe:

*

*you, and

*an object at the end of a string that you're swinging in a circle.

If you let go of the string, the object flies off in a straight line, travelling away from you at a constant velocity. Newton's first law says that an object that's travelling at a constant velocity experiences no (net) force: after you've let go, there aren't any forces on the object. If you're still holding onto the string, the object would be travelling away from you – but something's stopping it: a force is opposing that motion (the tension in the string, from you holding onto the end).
In what direction do you have to pull an object to stop it flying outwards?
Newton's second law says that, if there's a (net) force on an object, the object's accelerating in the same direction as the force, so the acceleration must be in the same direction as your pulling.

But why does the object keep going at the same speed, if it's constantly accelerating? Well, for the same reason that your car accelerates when you press the accelerator, then accelerates (in the opposite direction – also known as “deceleration”) when you press the brake, but doesn't have to keep getting faster forever.

*

*If acceleration is in the same direction as motion, you get faster.

*If acceleration is in the opposite direction to motion, you get slower.

*If acceleration is completely sideways to motion, you don't get faster or slower; you just change direction without changing speed.

(If you want to be fancy, you can split all different directions of acceleration up into forwards / backwardsness and sidewaysness, and work out how much your speed changes and how much you change direction, but that isn't necessary for understanding this.)
If the acceleration is always sideways (perpendicular) to motion, then the object will just keep changing direction without speeding up or slowing down. And if you draw a diagram, you'll see that the inwards / outwards line is always sideways compared to the outside of the circle; if you keep pulling towards the circle, the object will keep going 'round it.
A: Trying the shortest possible answer:
The ball is not a rocket. It has no mechanism to accelerate on its own, that is, it cannot change its own velocity. Therefore, the ball cannot accelerate once it is released. The ball flies straight away (Newtown's first law).
The mechanism by which it changes its velocity is obviously the rope, providing an external force.
It's the same as pulling a heavy block with a rope. You'll feel a counter-force (stiction force; centripetal force for the rotating ball), but the resulting acceleration is towards you.
A: If there were no force, the object would move along in a straight line along the tangent. But since that is not happening and the object is moving in a circle, there must be a force acting inwards that is constantly changing its direction. This is called a centripetal force.
A: There are some detailed explanations and some really good discussions here, but the confusion about the direction of acceleration has a very simple and short answer: it depends on the reference frame. You must specify which reference frame you're in while defining your acceleration. If you're standing on the ground and look at the spinning ball, then the acceleration is inwards (centripital) but if you were to choose the ball as your reference frame, then direction of acceleration flips (centrifugal).
You see, Newton's laws only work in an inertial reference frame (a frame of reference that isn't accelerating). The ground is (very much) an inertial reference frame, but the spinning ball definitely isn't.
In the reference frame of the ball, you must introduce a pseudo-force that is opposite in direction but equal in magnitude to the actual force (the string pulling the ball inwards). So, in that non-inertial reference frame (ball's), the acceleration is outwards.
Here's another classic example to make the idea rock-solid: if you're in a rocket in space and that rocket is accelerating upwards with an acceleration a. When you're inside the rocket, you'll feel as if something is pulling you downwards. But if someone is looking at you from outside the rocket, they'll tell you that no, the rocket it moving upwards and that's what is pushing against you.
Do you see it here as well? Your reference frame (inside the rocket) is non-inertial, so you conclude that there's this magical force which is pulling you downwards, so the acceleration must be down as well. But someone floating outside (inertial reference frame) will conclude the exact opposite. You're clearly accelerating upwards from his point of view.
A: I would explain the correct answer without reference to forces. Basically, this is a question about acceleration and I would not introduce forces or another reference system. in addition to the one where the motion is described as a circular motion.
The very simple kinematic fact is that the acceleration vector at a given time $t$ is defined as the derivative of the velocity at the same time $t$. If one would like to avoid derivatives, it is enough to analyze the average acceleration over a small interval of time $\Delta t$. Provided $\Delta t$ is small enough that the value of the average acceleration $\vec{a}_m=\frac{{\vec v}(t+\Delta t) - \vec{v}(t)}{\Delta t}$ does not change significantly for any smaller interval of time, this average acceleration can be used as the acceleration $\vec{a}(t)$.
Now, in a circular motion (uniform or not, does not matter), the velocities at two times $t$ and $t+\Delta t$ are not aligned (the velocity is always tangent to the circle). Moreover, whatever is the direction of $\vec{v}(t)$, $\vec{v}(t+\Delta t)$ bends toward the side of the trajectory where the center of the circle is.
The following picture shows the geometry
In particular, the difference vector ${\vec v}(t+\Delta t) - \vec{v}(t)$ has the tail on the tip of the vector $\vec{v}(t)$ and its tip on the tip of the vector ${\vec v}(t+\Delta t)$ (parallelogram rule). It should be clear that it is impossible to have an acceleration pointing in the direction opposite to the direction where the trajectory bends.
PostScriptum for more formal readers:
Of course, the previous elementary argument can be made completely formal by using a little of differential geometries of curves in 2 and 3 dimensions.
A: Let's consider an everyday example: Driving a car or a bike. If you drive on a straight line at constant speed you do not experience any force. That's boring (not part of your question), so let's drive in a circle. If we drive in a circle in the counter-clock-wise direction, we are constantly turning to the left.

However, in order to move to the left we must experience a force, which is pushing/pulling us to the left. Hence, taking this perspective it becomes clear that the force we are experiencing must be directed inwards, to the center of the circle.
The situation in reversed if we take the perspective of being the inwards pulling force. So if we have a mass on a string and we rotate it in a circle, the mass becomes the car/bike of the former story and we take the role  of the inwards pulling force. Since the mass experience an inwards pulling force, and since any force must be balanced (see Newtons law), we must experience an outwards pushing force.
Hence, whether we experience a force with is inwards or outwards directed depends on the role we play. Hope this helps.
A: A moving object continues in a straight line unless a force is applied to it.
If a ball is whirled in a circle  at the end of a string, it is caused  to move in a circle by the pull of the string.  If the string breaks the ball proceeds in a straight line unless gravity pulls it downward. The ball's straight line is a tangent to the circle. See the previous drawings showing that.
If there was a centrifugal force the released ball would move from its position directly away from the center of the circle like the symbol for Mars.  It does not do that.
