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The Theory of Relativity
Einstein's Theory of Relativity functions on these premises:
The formulation of General Relativity came about by a thought experiment by Einstein. Say someone was riding in an elevator, and a beam of light was coming through a window of the elevator. While the elevator was stationary, the light would appear travel in a straight line to the observer. However, if the elevator were to accelerate - say upwards - then the beam of light would appear to bend down towards the floor. With this thought experiment, Einstein concluded that acceleration could have an affect on the path of light. To the observer in the elevator, if the elevator were to accelerate at 9.8 meters / second2, he would feel the same "force" that gravity exerts on him if he were standing on the surface of the Earth (an apple falling from a table onto the floor falls at 9.8 meters / second2). These two ideas combined led to Einstein's "Principle of Equivalence". The Equivalence Principle states that acceleration can exert the same force on a body as gravity. Another example: pilots are put under G(ravity) Forces while flying; their acceleration puts the same force on them as gravity would. 2 G's of acceleration means the force is equivalent to 2 times the force of gravity.
The Equivalence Principle became a working hypothesis which needed to be verified. Verification came from a solar eclipse in 1919. Since the sun is a super-massive object, according to Einstein's Equivalence Principle, the sun's gravity should bend light much like acceleration in the hypothetical elevator. Because the solar eclipse allowed scientists to look directly at the sun and its immediate surrounding area, they could use this solar eclipse to validate Einstein's hypothesis. And they did. Stars that were behind the sun that shouldn't have been visible indeed were; the sun's gravity was bending light around it. The verification of this phenomenon provided positive evidence for Einstein's General Relativity, and also made him famous all over the world.
When Kepler proposed his 3 laws, they correctly predicted the movement of celestial bodies. Newton expanded on Kepler's formulas and devised equations for gravity. These equations predicted the movement of planets around the sun - except for Mercury. Careful observations of Mercury showed that the actual value of the precession disagreed with that calculated from Newton's theory by 43 seconds of arc per century. In general relativity, this orbit will precess, or change orientation within its plane, due to gravitation being mediated by the curvature of spacetime. Since the orientation of an orbit is usually given by the position of its periapsis, this change of orientation is described as being a precession in the periapsis of an object. However, the problem was resolved by Einstein's theory, which predicted exactly the observed amount of perihelion shift. This was a powerful factor motivating the adoption of Einstein's theory.
Thus, the predictions of general relativity perfectly account for the missing precession (the remaining discrepancy is within observational error). All other planets experience perihelion shifts as well, but, since they are further away from the Sun and have lower speeds, their shifts are lower and harder to observe. For example, the perihelion shift of Earth's orbit due to general relativity effects is about 5 seconds of arc per century. The periapsis shift has also been observed with Radio telescope measurements of Binary pulsar systems, again confirming general relativity.
Gravity In General Relativity
Einstein's theory of General Relativity superceded Newton's Laws of Gravity by being able to explain gravity more consistently. So what is gravity in General Relativity? In this model, gravity is the bending of space-time by massive objects, much like a bowling ball sitting on a bed. If the Sun is represented by the bowling ball, space-time would be represented by the mattress of the bed. In the 1919 solar eclipse, if light from a star behind the Sun can be represented by a moving object on the bed like a penny, the penny's path along the mattress would be distorted by the depression caused by the bowling ball. Instead of moving in a straight line, the penny would warp around the indentation caused by the bowling ball.
So, the mass of the Earth makes objects fall at a rate of 9.8 m/s2. The speed of light is 3 x 108 m/s. If gravity has an affect on the course of a beam of light, what happens if the gravity of an object makes objects fall towards it at a speed greater than 3 x 108 m/s2 (the speed of light)? The result is an object that reflects no light, since all light that enters its gravitational field doesn't have the speed to escape its gravity: what's called a black hole. While black holes have never been observed directly, there is plenty of indirect evidence of their existence. One such evidence is such that when astronomers look into the stars, they see intense warping of light coming from certain sources. Much like looking at a mirror that is warped from being melted in the center. Away from the mirror's warping, things look normal. As you head towards the warping, your reflection gets skewed. While we can't actually see the warping itself, we do see its effects.
Another evidence for black holes is the existence of Hawking Radiation. Hawiking Radiation is the strange emittence of high powered X-ray radiation coming from "black" sources - sources where no light enters or escapes. To explain Hawking Radation, some knowledge of quantum physics (QM) is needed. In QM, all particles have their anti-particle. QM states that virtual particles are created in the "vaccuum" of space (space actually isn't "empty") - these are called vaccuum fluctuations. A particle and its anti-particle will "poof" into existence due to quantum potentiality, but in order to conserve the total mass of the Universe, these two particles immediately destroy each other. If one of these vaccuum fluctuations happens near a black hole and one half of the particle/antiparticle pair is sucked into the black hole before the two particles annihilate each other, the black hole will seem to "emit" this particle with a lot of energy - energy equivalent to x-ray radiation, which we can see.
The existence of black holes can be considered a "prediction" of General Relativity, since it is a logical conclusion of the premises of General Relativity.
Global Positioning Satellites
GPS observation processing must also compensate for another relativistic effect, the Sagnac effect. The GPS time scale is defined in an inertial system but observations are processed in an Earth-centered, Earth-fixed (co-rotating) system, a system in which simultaneity is not uniquely defined. The Lorentz transformation between the two systems modifies the signal run time, a correction having opposite algebraic signs for satellites in the Eastern and Western celestial hemispheres. Ignoring this effect will produce an east-west error on the order of hundreds of nanoseconds, or tens of meters in position.  The atomic clocks on board the GPS satellites are precisely tuned, making the system a practical engineering application of the scientific theory of relativity in a real-world system.
So What IS Gravity?
The Theory of Relativity, like all scientific theories, is an explanation and model of natural phenomena. The Theory of Relativity provides a highly predictive model of how gravity works in many different contexts. However, what actually causes gravity is still left unexplained. There are many hypotheses as to what causes gravity. The other three fundamental forces (electromagnetism, strong and weak nuclear force) can be explained by their quantum physical counterpart (photons, nucleons/gluons, and vector bosons respectively). No such quantum counterpart exists for gravity. The "graviton" has been proposed, but no evidence has been presented to demonstrate its existence. However, Loop Quantum Gravity might be a promising hypothesis. Along with String Theory and Casual Dynamic Triangulation, these make up the most promising prospects for a theory of quantum gravity.
The stakes are high. If we can find out how gravity can be represented and function in a quantum environment, we might be able to build a model of what happened before the Big Bang. But, because of our macroscopic bias:
Quantum Physics prevents us from seeing "past" the singularity of the Big Bang.