Why is curved space able to change an object's velocity (vector)? I don't really understand what is meant by curved space. Why does mass warp space? Why does curved space alter the velocity of a massive object? Normally to change an object's direction you have to apply some force to overcome inertia. So how does curved space do it? What is space anyway?  Layman's terms, please.
 A: Ultimately, the reason we use curvature to describe gravity is because that's what fits with our observations. General relativity is extremely good at predicting the motion of objects under gravitational interactions. If you want to get some intuition for why we choose curvature to describe gravity, the reasoning is basically as follows.
Imagine that you are in a free-falling elevator. You will feel as if you are weightless. There is no way you could tell whether you are falling in a gravitational field, or whether you are in the depths of space, far from any source of gravity. Any experiment you can do will have the same results in both cases.
Actually, there is only one way to detect the presence of gravity -- if you could look out the window and see another elevator falling beside you, you would notice the elevator slowly coming closer to you. This is because both elevators are being drawn toward the center of the earth. In the absence of a gravitational field, any two objects that are "free-falling" (have no forces acting on them) will NOT be drawn closer to each other. In a gravitational field, two objects that are free-falling (have no forces other than gravity acting on them), may be drawn closer together. This is called a tidal effect, and it is gravity's only effect.
A convenient way to describe gravity mathematically is to use the mathematics of curved space-time. The basic rules are simple. Matter (and energy and all that) bend space-time. Objects always "think" they are travelling on a straight path, just like you might think that you are walking in a straight line on the surface of the earth. However, if you look at someone else travelling beside you in a "straight" path, even if you start out moving away from each other, the curvature may eventually bring you back together (on the surface of the earth, you'd have to walk pretty far for this to happen, but it could happen). If you are unaware of the curvature, it might look as though you are being pulled together. That's how gravity works.
Some objections:
This may seem very counter-intuitive. On the surface of the earth, we think that we feel the force of gravity. Actually, though, the force that we feel is the force of the ground preventing us from continuing on our natural, "straight" path.
It can be hard to reconcile this geometrical point of view with everyday experience. As an example, consider two massive balls, at rest with respect to each other in empty space. It can be hard to understand why curvature would cause these two balls to come together, since they are not moving through space. The key thing, however, is that they ARE moving through time. In relativity, space and time are not two separate concepts. Gravity curves both space and time, so as the balls move through time, the gravitational curvature will cause them to bend into each other.
A: You must have heard the phrase "matter tells spacetime how to curve, and spacetime tells matter how to move".
Now in reality, it is not mass, but stress-energy that causes spacetime curvature. Anything that does have stress-energy (and currently any elementary particle we know about) does have stress-energy and does curve spacetime.
If you would like an analogy, spacetime itself is the tracks for a train, the train cannot go another way, it must follow the (curvature) of the tracks.
Now imagine there is a little curvature on the tracks, maybe the tracks bend very little over a 100 miles, but if you are on the train, the tracks seem just straight, you do not notice the curvature locally. This is how in our normal everyday life curvature is, you can only notice local spacetime curvature (other then the fact that the ground is accelerating upwards) if you go close to a extreme object, like a black hole.
Now you are asking how does spacetime do it? How does it change the direction of an object without a force acting on it? Just like the train must follow the tracks curvature, any object (that we know of currently, meaning any elementary particle) must follow spacetime curvature, this is what we see from all experiments.

Suppose I'm orbiting the Earth. The spacetime curvature is controlling my motion i.e. I move in a circle centred on the Earth rather than a straight line because the spacetime in my vicinity is curved. This is an example of Wheeler's statement - the mass of the Earth curves spacetime and the curvature tells me how to move.
there is an important distinction between acceleration due to an applied force and acceleration due to spacetime curvature. If I'm floating in space then I can let go of an object and it will remain floating next to me. This applies whether I'm orbiting the Earth or whether I'm floating in empty space far from any masses. My acceleration relative to a released object is called the proper acceleration and it's an important invariant in relativity. Any object that is moving solely in response to spacetime curvature has a proper acceleration of zero.

"Spacetime tells matter how to move; matter tells spacetime how to curve" and acceleration in flat space-time?
We, and all objects we currently know of, exist in spacetime, and must follow its curvature.
A: Actually, for non-relativistic velocities, it's the curvature of time (being a part of the curved spacetime) that changes the velocity of a massive object near the Earth. Because time runs slower the closer you are to Earth, the velocity of a massive object will change to maximize the time passed in the frame of the object (called the proper time).
Usually, the change of a massive object's velocity is shown by taking a stretched sheet of flexible rubber and then putting a massive object in the middle, because of which the rubber sheet will bend and is said to represent the curvature of space. See, for example, this demonstration. It seems that a small marble put on this sheet will change its velocity because the sheet is curved. But all this happens only because the real gravity is pulling on the marble (and on the heavy object that curves the sheet), and that's what makes the marble move (because the curvature of time, as part of the curved spacetime). So the false impression is given that the curvature of space is the reason.
For objects moving at relativistic velocities, it's a combination of both the curvature of space and the curvature of time which changes the velocity. In the case of light, it's only the curvature of space which makes the light move in a geodesic path, a path with the smallest distance in the curved space. Light doesn't move through time. Time stands still for a photon.
In flat space (where we can apply special relativity contrary to general relativity) we can compare this with the case of special relativity in which an on an object always moves through spacetime with the same velocity, the speed of light. Non-moving objects only travel through time. Objects moving with a non-zero velocity move both through space and time. Massless objects move only through space. But the object moves always with the speed of light through the flat spacetime with the velocity of light.
How cause mass the curvature of space. I think it's just a fact of Nature, which can't be explained (not even in the context of a not yet found quantum gravity, which makes one doubt if it even exists). In general, space(time) without mass is flat, so I guess space(time) with mass has to be different from flat space, i.e. curved. Mass and space can't exist without each other.
A: As far as I know, it is not known why mass warps space (one of the biggest problems in physics, uniting general relativity with quantum mechanics). It is just a model, and all observations so far support this model. As for the second question, as far as I understand spacetime does not change an object's velocity, it just appears to change from an outside observer's viewpoint. That is, it is traveling straight and with constant speed through it's surrounding spacetime, but since this spacetime is curved locally, if you are watching from a distance (in a part of spacetime curved differently), it appears as if the object is accelerating or in a curved trajectory.
A: If you notice in GR, it always mentions space time is curved, it does not mention space is curved. When a heavy object curves space time, not only space component is affected, the time component is also affected. We all travel through time, even an absolute stationary object travels/ is moving in time domain. When a object  enters the curvature, the objects flat time domain enters to streched time domain, for an outside observer (outside curvature) this change appears as beginning of motion( like a slip on slippery surface or a rate of change -dt). As you move closer to the heavy object the more space time is curved hence the more rate of change you see.. hence the illusion of acceleration due to gravity appears.
Let's consider this example, an object is moving 1 meter per second out side a curvature. Now the object enters a curvature where the space time is stretched  to 2meter at the starting of curvature then 3 meter and then 4 meter and so on... please note that the stretched 2,3,4 meter are equivalent to 1 meter outside curvature, now along with the space, time is also stretched, ie at 2 meter is 2 seconds, at 3 meter, it is 3 seconds so on etc... Hence the 1m/s moving object after entering curvature appears to accelerate for an outside observer, hence the illusion of falling. We actually need to apply force to stop this falling object to the center of curvature.
A: It is a postulate of general relativity that matter moves along the geodetics of curved space. Without this postulate it would not matter if you describe space with curved coordinates.
