Relativity deals with motion.  It is an important topic to discuss since motion is an integral part of life. Life persist when kinetic interplay with static.  Therefore relativity reveals the relationship between objects (static and dynamic)  as well as their energy of interaction. 

The main idea behind relativity is that the universe has 4 real dimensions.

Thus the usual distance between two points we know is not complete. The true distance must be between points in the 4 dimensions universe and take into account the time. From that very simple idea you find a number of interesting results. For example the magnetic field appears as a correction of the electric field, the energy and quantity of motion are part of the same physical vector, the kinetic energy is a correction of the mass energy etc...

The name "relativity" comes from the Galilean principle of relativity that is now interpreted in relativity as simply arising from the isotropy of the universe.

Theory of relativity is generally referred to Einsteins theory of relativity where time is also considered to be relative and velocity of light is a constant independent of the motion of the observer. When we deal with uniform relative motion between two frames, theory is known as special theory of relativity. However, in case of non uniform relative motion, theory is known as general theory of relativity. When relative velocity is very high near velocity of light, drastic change  is observed in time interval of two events, length of the body, mass of the body in other frame. More details can be obtained from any text book on relativity.

In terms of how it affects "general life", here's an example : relativistic effects are built into GPS calculations. If these effects were overlooked, GPS would be significantly less precise.

The idea of relativity is very old.  Consider Galileo's "Parable of the Ship", which gives an excellent physical description of what would be considered the indistinguishability of inertial frames.  Einstein demonstrated the invariance of Maxwell's equations in inertial and accelerated frames by developing his Special and General Theories of Relativity.

***Parable of the Ship***

SALVATIUS:

Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies, and other small flying animals.

Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it.

With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin.

The fish swim indifferently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need to throw it no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction.

When you have observed all of these things carefully (though there is no doubt that when the ship is standing still everything must happen this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that.

You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still.

In jumping, you will pass on the floor the same spaces as before, nor will you make larger jumps toward the stern than towards the prow even though the ship is moving quite rapidly, despite the fact that during the time that you are in the air the floor under you will be going in a direction opposite to your jump.

In throwing something to your companion, you will need no more force to get it to him whether he is in the direction of the bow or the stern, with yourself situated opposite.

The droplets will fall as before into the vessel beneath without dropping towards the stern, although while the drops are in the air the ship runs many spans.

The fish in the water will swim towards the front of their bowl with no more effort than toward the back, and will go with equal ease to bait placed anywhere around the edges of the bowl.

Finally the butterflies and flies will continue their flights indifferently toward every side, nor will it ever happen that they are concentrated toward the stern, as if tired out from keeping up with the course of the ship, from which they will have been separated during long intervals by keeping themselves in the air....

SAGREDUS:

Although it did not occur to me to put these observations to the test when I was voyaging, I am sure that they would take place in the way you describe.

In confirmation of this I remember having often found myself in my cabin wondering whether the ship was moving or standing still; and sometimes at a whim which I have supposed it going one way when its motion was the opposite....

***Galileo Galilei, Dialogues Concerning the Two Chief World Systems (February 1632) This translation is by Stillman Drake (University of California Press, Berkeley, 1962, pp186-)

***In italiano: Dialogo sopra i due massimi sistemi del mondo, http://it.wikisource.org/wiki/Dialogo_sopra_i_due_massimi_sistemi_del_mondo_tolemaico_e_copernicano