The reason you may feel like you get different answers each time you ask is because there is a lot to explain in relativity and a few different ways to explain each thing. The best thing you can do is get a book on it:
http://math.ucr.edu/home/baez/physics/Administrivia/rel_booklist.html
There are two theories of relativity, the
special theory and the
general theory. The special theory was completed in 1905 as a remedy for the conflict between Maxwell's electromagnetism and Galilean mechanics. This conflict was really only something Einstein cared about, but his special theory of relativity also happened to explain something else physicists were confused about at the same time, the results of the Michelson-Morely experiment which seemed to show that the speed of light is the same to everyone, no matter how fast they're moving. Einstein
assumed the results of this experiment (that the speed of light is a constant) on the basis of the principle of relativity (an expanded version of a Galilean postulate) and Maxwell's theory of electromagnetism, and ended up with a new system of mechanics which reduces to Galilean mechanics when velocities are low relative to the speed of light.
The general theory was completed in 1916 as a remedy for the conflict between the special theory of relativity and Newton's theory of gravity. The general theory also solved an experimental problem physicists were confused about at the time, although not quite as accidentally, the perihelion shift in the orbit of Mercury. Newton's theory of gravity doesn't predict the orbit of Mercury around the Sun accurately, but Einstein's general theory of relativity (the replacement of Newtonian gravity) does give accurate predictions for the orbit of Mercury.
Some unfamiliar/weird insights from relativity:
(1) Something moving fast will be shorter than it would be if it is standing still.
(2) A clock moving fast will tick slower than it would if it was standing still.
(3) Two events which happen at the same time from one person's point of view can happen at different times from someone else's point of view (provided the two people are moving fast relative to each other).
(3) Matter and energy are united as two different aspects of the same thing.
(4) We live in a four dimensional universe; space and time are united as the spacetime continuum.
(5) In a strong gravitational field, lengths contract and time slows down.
(6) Gravity isn't a "force" in general relativity; it is a curvature in spacetime, which violates Euclid's parallel postulate (allows two inertial bodies moving parallel to each other to fall toward each other).
(7) Nothing can ever accelerate to a faster speed than the speed of light.