In general relativity, an observer in freefall and an observer standing still on a body with mass such as the Earth see no difference in the movement of an object they drop. This is known as the equivalence principle.
In order to understand General Relativity, imagine a suspended, stretchy table cloth. It is perfectly flat. Now imagine a ball, and drop it onto the table cloth. The table cloth is very big, so the ball cannot drag the cloth with it. Instead, the ball pushes on the cloth and makes a valley. There is no hole. The curved cloth around the ball is its gravity. If an object enters the valley, it will fall towards the bottom, where the ball is. If the object moves sideways when it enters the valley, then it will move both sideways and towards the ball. This creates an orbit around the ball.
The Sun can be seen as this kind of valley in spacetime, and one of the other objects in the valley is the Earth. The Earth does not roll directly towards the Sun (or ball) because it is moving too fast. The force pulling the Earth towards the sun is about the same as a second force. This second force is called the centrifugal force. The centrifugal force exists because the Earth moves sideways. This sideways motion makes the distance between the Earth and Sun increase. Since the Earth is being pulled towards the sun and moving away at the same time, it stays at about the same distance. This is also how the Moon orbits the earth. In this second case, Earth is the ball and the Moon is the object.
General relativity has predicted many things which were later seen. These include:
As light gets closer to the sun, it bends towards the sun twice as much as classical physics (the system used before general relativity) predicts. This was seen in an experiment Arthur Eddington did in 1919. When scientists saw his experiment they started to take general relativity seriously.
The perihelion of the planet Mercury rotates along its orbit more than is expected under Newtonian physics. General relativity accounts for the difference between what is seen and what is expected without it.
Redshift from gravity. When light moves away from an object with gravity (moving away from the center of the valley), it is stretched into longer wavelengths. This was confirmed by the Pound-Rebka experiment.
The Shapiro delay. Light appears to slow down when it passes close to a massive object. This was first seen in the 1960s by space probes headed towards the planet Venus.
↑A non-mathematical account is: Einstein, Albert and Infeld, Leopold 1938. The evolution of physics: from early concepts to relativity and quanta. Cambridge University Press. Reprinted 1966, Simon & Schuster, New York. ISBN 0-671-20156-5