As a geometric theory of gravity, General Relativity combines Einstein's Special Relativity with Newton's law of gravity. Essentially, it describes gravity as the geometry of spacetime. The equivalence of gravity and acceleration leads to time dilation caused by gravity; for example, time runs more slowly on the sun's surface by 2 parts in 1 million; that is, approximately one second every 5.8 days. There is also the prediction that gravity bends light. While Newton also predicted this affect, it anticipated only half the amount of shift predicted by General Relativity. Redshift caused by gravity was another prediction. All of these predictions have been observed experimentally. For example, Sir Arthur Eddington proved the bending of light in a gravitational field during a total eclipse way back in 1919. He showed that stars near the Sun appear to be slightly shifted because its gravitational field has curved their light. Eddington's results agreed with Einstein's prediction. This effect leads to gravitational lensing where multiple images of a distant object, for example, a galaxy, are seen due to a large intermediate object bending light.
The "Einstein Cross" shown on the right is a gravitationally lensed quasar that sits directly behind the galaxy "Huchra's Lens". It produces four distinct images forming a cross, with Huchra's Lens itself at the center of the "cross". The quasar is about 2.5 Gpc away, and the Huchra's Lens galaxy is about 120 Mpc away. Remember, the edge of the observable Universe is only 4.2 Gpc away. General relativity also explained the anomaly in the orbit of the planet Mercury, described in the introduction, that had been found in 1859. Based on Newtonian physics, the calculated perihelion precession fell short of the observed value by approximately 43 arc seconds each century. This was the same amount predicted by General Relativity. In the case of the Earth, the adjustment is 5 arc seconds per century. The effect is also observed, much magnified, in binary pulsar systems; for example, in J0737-3039 (see below) the relativistic perihelion precession is a massive 16.9 degrees.
General Relativity also makes predictions about how the generation of gravity waves by massive bodies causes them to lose energy. As an example, the binary pulsar J0737-3039 comprises two pulsars (J0737-3039A and J0737-3039B) that orbit each other at a distance of only 800,000 km (500,000 miles), once every 2.4 hours. The system is between 1,600 and 2,000 light-years away. General Relativity predicts that the production of gravitational waves by the two stars, about 1.33 and 1.25 solar masses respectively, but only about 10 km (~6 miles) across, will cause them to lose orbital energy and move closer together. In the case of J0737-3039, General Relativity predicts that they will reduce the size of their orbit by 7 millimeters each day; that is by more than 2.5 meters (8 feet) a year. A number of independent radio observations from around the world have confirmed this prediction. Over time, the effect will increase the rate at which they approach, and they will coalesce in about 85 million years, just possibly forming a black hole, depending on how much material is thrown off during the merger.
Karl Schwarzschild provided the first exact solution to Einstein's field equations, which included the first hint of the possibility of Black Holes. While pleased that a solution had been found, Einstein refused to believe in the possibility that Black Holes would be allowed. In any event, the Schwarzschild Black Hole was a purely theoretical object, and would not exist in nature as it had no spin or charge, and had to be the only mass in the Universe. Subsequently many black holes have been identified, including supermassive ones at the centers of most, if not all, galaxies including our own Milky Way.
Interestingly, the clocks in orbit for Global Positioning Systems (GPS) are adjusted to take account of relativistic effects due both to Special Relativity, the satellite's speed, and General Relativity, the difference in gravity on Earth and in orbit. Without adjusting for relativity, GPS errors would accumulate at around 7 miles per day! To adjust for this, clocks on the satellites tick 38 millionths of a second faster per day than clocks on Earth or, more particularly, clocks in the GPS receivers. In addition, satellite orbits are not perfectly circular, so corrections are also made for the elliptical orbits.
The UCLA site has a nice tutorial on relativity, without too much math. The University of Illinois has a description of General Relativity.