Others have already given you the gist of it.
What I find really interesting is how both types of time dilation (and most of the rest of both special and general relativity) follow from three simple observations about the world.
1. Imagine you are in a spaceship that is deep in interstellar space. Your engines are off and you feel no acceleration. There is another ship in the direction you consider to be behind your ship, moving in a constant velocity in the direction you consider to be the front of your ship. The people in the other ship also have their engines off and feel no acceleration. From their point of view, they think you are in front of them, moving backward at constant velocity.
The first observation that forms the basis of relativity is that neither you or the people in the other ship can do any physics experiment to determine which if you is "really" moving.
2. The second observations is that the speed of light is the same for all observers.
Suppose you make a clock that consists of a source the emits a very brief pulse of light, perpendicular to the direction the ship is pointing. This pulse bounces of a mirror that you have placed 0.5 light-nanosecond away from the emitter. When the pulse reflects back and reaches the emitter, the emitter emits another pulse. Every time a reflected pulse gets back to emitter counts as a tick of this clock, so it ticks 10^9 times per second.
Suppose the people on the other ship also make such a clock, and as your ships pass you synchronize your clock with theirs.
As the ships pull apart and you watch their clock and yours, you will note that because their shipping is moving, their light is not moving perpendicular to the direction the ships are pointing like yours is. Theirs has to move diagonally so it hits the mirror where the mirror will be after the light cross that 0.5 light-nanosecond gap between the emitter and mirror.
Now if this were a bouncing ball clock instead of a light clock, that would be no problem. You would see their balls as having the same perpendicular velocity as yours, plus their would have a forward velocity equal to the speed you see their ship moving, and so the diagonal velocity would combine both of those to something bigger than either (for the same reason the hypotenuse of a right triangle is longer than either side). Their ball would have farther to go than yours, but it would be going faster, so their clock would tick at the same rate as yours, and all would be as common sense requires.
But light has the same speed for all observers! So you do not see their light going faster than your light. It's going the same speed, but has farther to go because of the diagonal. So you see their clock ticking slower than your clock!
But #1 says that we can't tell who is "really" moving. That means that every clock the other ship has must slow down by the same rate their light clock slows down or they could tell their are moving because their light clock goes out of sync with their other clocks.
But when every possible way to measure time slows down at the same rate, it is hard not to say that time has slowed down.
Keep going on in this direction, reconciling the speed of light being constant for all observers and the impossibility of determining which ship is "really" moving, and you end up with pretty much all of special relativity--the time dilation, the Lorentz contraction, the Lorentz transformation, and so on.
3. The third observation is that if you turn your engines on and they provide uniform acceleration, it looks like gravity to you. If your windows are closed you can't tell if you are accelerating or if you ship is at rest on top of a planet. This holds even for non-uniform acceleration--you cannot tell if the forces you feel are due to varying acceleration or varying gravity.
Consider two people in a large cylinder rotating around its axis, so they feel a force pushing them toward the wall. Suppose they each have a clock, and the clocks are synchronized. Now suppose one of the person takes his clock, climbs a ladder up to the axis of the cylinder, hangs out there a while, and then climbs back down.
You are watching that. You see that the guy who stays on the wall is moving fast, so you see his clock moving slow. As the other guy climbs, you see his clock speeding up, because you see him slowing down as he gets nearer the axis. When he's at the axis, his clock runs the same speed as yours. Then when he climbs back "down", his clock slows down more and more, until he gets back with his buddy. Their clocks are again running at the same speed, but they are now out of sync.
From your point of view, this is all just special relativity, like we worked out with the two spaceships.
But for the two guys in the cylinder, #3 says they need to get the same result if that force toward the wall is due to gravity, not acceleration. From their point of view they are in a gravitation field pointing toward the wall, and the guy on the ladder climbed up and then back down. So we have to conclude from this that clocks run faster when you are "higher" in a gravitational field.
If you imagine now that they two guys decide to measure Pi, by measuring the circumference and radius of the cylinder, you can similarly apply special relativity from your point of view to figure out what they will measure, and find that they get a value of Pi that is too high. By #3 that means the same thing must happen in a gravitational field, and from that we (well...if we were Einstein) would be able to figure out that this means mass must warp space.
There are some books meant for the general public that go over this far better than I just did. This is getting long, so I'm going to post the book recommendations in a reply to this.