Einstein developed two theories of relativity, the first of which is a subset of the second. In the Special Theory of Relativity, Einstein hypothesized (1) that the laws of physics (electromagnetism in particular) would not depend upon how fast a person was moving, and that (2) the speed of light would been seen to stay the same that no matter what velocity a person traveled at. As a result, there could be no absolute measurement of velocity, only relative measurements. When formulating this mathematically, this hypothesis implied that space and time were related, and that mass and energy were equivalent and could be converted between each other.
In General Relativity, Einstein hypothesizes that gravitational attraction and acceleration were equivalent. This meant that any time a force acted on an object, it could be thought of mathematically as undergoing an acceleration. When formulated mathematically, this theory implied that mass (what we think of as both matter and energy), which is the source of gravity, caused space to be curved. An object traveling through this curved space would appear to accelerate in the direction of the gravitational force.
Whereas Special Relativity dealt with objects moving at a constant velocity, General Relativity was developed to incorporate gravity and acceleration into Einstein's theories. In this sense, it is one of the few successful scientific theories that was motivated in large part by philosophical and mathematical considerations. However, once the mathematical framework of the theory was developed, Einstein quickly realized that it could explain one outstanding problem --- the fact that the orbit of Mercury precessed faster than would be expected according to Newton's theory for gravity.
Both theories of relativity have provided an enormous number of predictions that have been verified through experiments and observations. Special relativity predicts that a moving object should appear shorter than when it is standing still; a clock that is moved will appear to run slower than one that is left standing still (for instance, unstable but fast-moving particles will appear to last longer than similar particles that are at rest); and that two events that occur at the same time to someone standing still will appear to happen at different times to someone moving.
Special relativity also predicted that mass and energy was equivalent, which was dramatically confirmed with the development of nuclear fission (heavy atoms splitting into two smaller ones) or fusion (to light atoms combining). When light atoms fuse or heavy atoms split, some of their mass can be observed to be converted to energy.
Finally, special relativity predicts that as an object is accelerated, it will approach the speed of light, but never reach it. This is commonly seen in particle physics experiments.
General relativity made Einstein famous when he successfully predicted the amount by which light should be bent when it passed by a heavy object. This was first observed during a solar eclipse, by measuring how a star appeared to move as it passed near the limb of the Sun. In the last 20 years, gravitational lensing has been observed as nearby stars pass in front of distant ones, and in cases where collections of galaxies lie along the same line of sight.
General relativity also predicts that clocks will run slower when they are in stronger gravitational fields. This has been confirmed by comparing the rate at which atomic clocks run at sea level and at high elevations. Similar effects can be seen in astronomical pulsars that are in close orbits with white dwarfs.
As mentioned above, special relativity predicts that mass can be converted to energy (and vise-versa). This principle underlies the development of nuclear technologies. The energy released when splitting heavy atoms can be used to power nuclear reactors. The energy released when fusing light atoms or splitting heavy ones can be used to power bombs (H-bombs and fission bombs respectively).
General Relativity is important to technology because it turns out that it is necessary to calculate the rate at which clocks run at different altitudes in order for the Global Positioning System to work.
Special relativity is intimately related to electromagnetism, because they share much of the same mathematical framework. Indeed, Special Relativity was developed so that it would reproduce invariant aspects of electromagnetic phenomena. Special relativity then served as one basis for the development of quantum electrodynamics by Dirac and others, which extends electromagnetism to quantum phenomena.
General relativity is widely used to explain astrophysical phenomena, particularly neutron stars (pulsars) and black holes (see below). General relativity is also used to describe the structure of the Universe, and is therefore fundamental to the development of the Big Bang theory.
Special relativity is in some sense complete, but only because it is limited in scope to objects moving at a constant velocity (the same could be said for Newton's theory of gravity).
We know that General Relativity has to be wrong on some level, because it is inconsistent with the other major 20th century physical theory, Quantum Mechanics. Crudely speaking, this is because space is smooth in General Relativity, whereas it must be broken up into small chunks (the Plank length) to accommodate the uncertainty principle in Quantum Mechanics.
Moreover, all of the test so far of General Relativity involve its weakest effects. These effects --- the bending of light, the changes in the rate at which time flows that depend on altitude, the deviation from Newton's gravitational potential --- are striking, but they don't serve as confirmation that the entire mathematical framework is a good description of the Universe. Therefore, physicists are searching for ways to confirm even more dramatic predictions of Einstein's theory, which requires studying effects of the strongest gravitational fields that Nature can produce. One thing they are looking for are gravitational waves, which are ripples in space and time sent out when very massive objects are disturbed (such as the core of a collapsing star) or collide.
Astronomers are also trying to prove the most startling prediction of General Relativity, that objects can get so massive that they cut themselves off from the rest of the Universe, forming a black hole. Astronomers believe they have found black holes that formed when massive stars collapse, and monstrous black holes that lie in the centers of galaxies. They refer to these as black holes because, under any physical theory that we now know, the objects they've found are too dense to avoid cutting themselves off from the rest of the universe. What we need to prove is that these objects have "event horizons", which means that light that gets too close to one of these objects can never get out. In some cases, there is circumstantial evidence* that event horizons exist. However, future experiments may be able to actually make an image of the blank spot in space where the event horizon of the black hole in the center of our Milky Way lies.
* Note: My own opinion is that the linked result demonstrates some lack of imagination as to what else could mimic the appearance of an event horizon. But I digress.
Like science itself, these pages are under construction. OK, so they are in a lot worse shape than science. I welcome your comments.
Wikipedia has more comprehensive discussions of the topics above.