Lay explanation of the special theory of relativity? What is Einstein's theory of Special relativity, in terms a lay person can follow?
 A: Special Relativity derives from two basic ideas:


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*The speed of light (in a vacuum) is always c.

*The laws of physics are the same in all inertial reference frames (basically, points of view that aren't accelerating, that is, they obey Newton's Laws.)


With these two points and a little math, various proven conclusions may be derived:


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*Time Dilation: When something moves fast relative to something else, time for the faster moving body slows down. It's not an illusion of time slowing down, it's the real thing: individual atoms that make up the body operate slower, chemical reactions function slower, and biological processes (aging) occur slower. From the perspective of the faster moving body, its time progresses at the usual pace.

*Length Contraction: Objects moving fast relative to other objects shrink along the line of the direction they're moving.

*Relativistic Simultaneity: There's no such thing as simultaneous events: because time is attached to the observer, different people could witness 2 events happening in different order. The exception to this is "causally-related" events which are events where event A is the cause of event B.

*Mass-Energy: The math goes into describing the mass of bodies at rest and how that mass changes as the bodies move. As bodies speed up they get "heavier." Nothing with mass can travel faster than light (and nothing with mass can travel AT the speed of light) because any massive body would reach infinite "relative mass" at that speed. You can derive $E=mc^2$ and fission/fusion from this.


This is a very quick summary of the basic points and principles.
A: Special relativity is based upon the idea of how the same events for different observers are located by each observer using their own rulers and clocks; their own measurement of space and time. It's also based upon the principle that the laws of physics don't change for observers travelling at different velocities to one another.
The average lay person already has an intutive understanding of how they think events are seen to occur by observers travelling at different velocities to one another. It comes under the name of Galilean relativity after Galileo first introduced the idea.
For example, a person driving a car will press the clutch at some time t1, and then press the break pedal a time t2 later at approximately the same location in his space; a zero space interval. On the other hand, someone on the road observng the driver will think these two events occured with a different space interval not equal to zero in his space, but separated by the same time intervale t2-t1.
Special relativity instead proposes that the time inverval between events for different observers also changes in a way similar to that for space intervals. It also means that events being simultaneous with one another is relative, as Einstein emphasised in his 1905 paper.
A: An important point about relativity is that it is not quite like the physics you learn in an introductory physics class.  There, you learn about Newton's Laws or Snell's Law or Lenz's Law, etc.
Those are all laws that tell things how to act; they tell mass how to respond to force, or light how to bend, or currents which way to run.  
Relativity is different in that it provides a set of meta-laws, or laws that the other laws of physics must obey.  It doesn't directly tell things what to do.
The classic example is Maxwell's Equations.  These are laws that tell charged particles and electromagnetic fields how to act.  It turns out that these laws obey a certain mathematical criterion, called "Lorentz Invariance", that is required by relativity.  So Maxwell's equations are good relativistic laws.  They obey the meta-laws.
On the other hand, Newton's laws (of motion) are not good relativistic laws.  They don't obey the meta-laws.  So in relativity, we need a slightly new set of laws to describe how mass responds to force.
As for what the meta-laws are, they were outlined by Nick Gotch as "basic ideas" above.  Those basic ideas turn out to be equivalent to Lorentz invariance.
A: 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.
