First of all we need the concept of inertial frames of reference (often just abbreviated as "frames"). If something is not acted on by forces, and is therefore not accelerated, then it could be thought of as being at rest, or it could have any constant velocity, in any direction, at any speed. If we fix our attention on the object or observer in that situation, we see that there is no reason to ascribe any particular speed or direction. Such a situation is called an inertial frame of reference. Einstein postulated two things about inertial frames:
1. In every inertial frame physics is the same. Two different physicists might have some velocity difference between them, but as long as neither of them is accelerated, their experiments will come out the same as long as everything in the experiments is in the same frame, i.e. moving along together ("comoving").
2. The speed of light is the same in all inertial frames. This is the important difference from previous physics. The naive addition of velocities would say that if Alice measures the speed of light to be c, and Bob is moving with velocity v with respect to Alice, then Bob, at least in Alice's view, will measure the speed of light as v+c. But not so, Bob only measures c, and Alice agrees that he measures c.
Putting these postulates together, Einstein showed that when Alice and Bob compare notes (which they can only do by sending signals, which obey the two postulates), each will see the other's lengths shrunk by a factor that depends on the speed between them, and each other's times strung out by the reciprocal of the same factor. This is not an illusion, the only real physics Alice and Bob can do together is via signals, and this means the results they get are as real as it gets; but it's relative to the speed difference between them.
Now imagine that Alice and Bob are fraternal twins. Because for each of them the other's clocks tick slower, each will see the other as aging more slowly than they do. And as long as their frames of reference remain inertial, this is symmetric; each of them sees the same effect in the other.
But imagine that Alice remains inertial, while Bob accelerates away and then back. This breaks the symmetry; Bob won't see the same thing Alice does. And it works out that in this situation; with Bob first accelerating away, then turning around and deceleratiing back, that when the twins are again comoving in the same inertial frame, then Bob will have aged less than Alice. This is the twin phenomenon. It is sometimes called a paradox, but it it straightforward physics, just counter to our naive expectations.
Is this enough to snow your teacher? I left out all the math because even Einstein couldn't have learned all that under the gun.
