Note: I will assume that LaTeX support will be added soon, so ignore my forumlas for now.
Einstein's Special Theory of Relativity cannot be of much significance to a lay person because he has no use for such knowledge. However, the common idea that this topic is extremely complicated motivates the layman to learn it.
The basic concepts of this theory are actually somewhat simple. In short, when an object is moving a certain velocity $u$, a few seemingly unusual phenomena happen to it. If a bar of length $L_0$ moves in the direction of its length, its new length will appear to be $L_0 \sqrt{1 - u^2 / c^2}$ from the our viewpoint on the ground, where $c$ is the speed of light. If that formula means nothing to you, simply take note that the length decreases as the velocity increases. Of course, from the bar's viewpoint, it will appear that our length has changed by that factor.
Since time is proportional to distance traveled at a constant velocity, the length contraction can show that the "rate of time" also changes when an object is moving. If a clock is traveling through space close to the speed of light, it will tick significantly slower than it did at rest.
Those are the basics, but phenomena such as mass-energy equivalence and nuclear binding energy can be derived from these concepts and experiments conducted in the early 20th century.