There have been numerous tests of relativity over the more than a century since Einstein published his first paper on the subject. The most complex tests are those for small general relativistic effects, using our solar system as a "test-bench".
The three major tests of relativity touched upon in this page and its attachments are: the perihelion shift of Mercury, the deflection of light by the Sun and the Shapiro time delay of light. Then there is the very contemporary Gravity Probe B, the measurement of the geodetic effect and inertial frame dragging.
The perihelion shift of Mercury
The anomalous perihelion shift of Mercury was a well-known fact by the time Einstein completed his general theory of relativity. After astronomers have compensated for all know effects of other planets on the orbit of Mercury, they were left with an inexplicable 43 arc-seconds per century.
Einstein's first "reality check" on his theory of general relativity was indeed to use an approximate solution to his field equations and calculate the perihelion shift of Mercury. It came out so close to the 43 arc-seconds per century that Einstein said it gave him "palpitations of the heart".
Deflection of light by gravity
Einstein also predicted that the Sun would deflect a close-passing (or "grazing") ray of light by about 1.75 arc-seconds. Sir Arthur Eddington measured the deflection in 1919, shortly after WWI. Although his measurement was not very accurate, it apparently confirmed the deflection as predicted by Einstein. This was the beginning of Einstein's fame.
The time delay of light
Light is "delayed" when it passes close by the Sun, or close by any other very massive object. Einstein did not predict this effect. It was Irvin Shapiro who first made the calculations and performed experiments to confirm the prediction. He bounced radar signals off Mars when it passed near to directly behind the Sun and measured the time.
The time it should take for the radar signals to make the return trip to Mars (without passing the Sun) is calculated using all available astronomical information, e.g., the positions of Earth and Mars relative to each other at any moment.
The difference between the calculated time and the actual measured time is the so-called Shapiro time delay of light. It is about 250 microseconds on a total round trip time of about 45 minutes.
The attached pdf file will give you an engineering view of the above tests of relativity.
Gravity probe B
Gravity Probe B is a spacecraft in a polar orbit about 650 km above Earth, intended to measure the properties of space-time around Earth. The two effects that it is designed to measure are the "geodetic effect" and "inertial frame dragging".
The geodetic effect is caused directly by space-time curvature. A gyroscope in space 'leans over' into the slope of space-time. When moved around earth in one orbit, the spin axis will point in a slightly different direction than the direction it started from.
Inertial frame dragging or 'gravitomagnetism' is caused by any rotating mass, which tends to drag the close-by space-time around with it, somewhat like a ball spinning in liquid does with the liquid. This effect alters the spin axis of the gyroscope in a different direction than the geodetic effect (if the gyro is in a polar orbit).
Gravity Probe Bís four gyroscopes were operating in "science mode" from about Sep 2004 to Sep 2005. The team is presently in a data analysis phase and the results are expected early in 2007.