Hundreds of years of observation have established the existence of a
universal attraction between physical objects.  In 1687, Isaac Newton
quantified this phenomenon in his law of gravity, which states that
every object in the Universe attracts every other object, with a force
between any two bodies that is proportional to the product of their
masses and inversely proportional to the square of the distance between
them.  If M and m are the two masses, r is the distance, and G is the
gravitational constant, we can write:
                     F = GMm/r^2 .
The gravitational constant G can be measured in the laboratory and has a
value of approximately 6.67x10^{-11} m^3/kg sec^2.  Newton's law of
gravity was one of the first great "unifications" of physics, explaining
both the force we experience on Earth (the fall of the proverbial apple)
and the force that causes the planets to orbit the Sun with a single,
simple rule.

Gravity is actually an extremely weak force.  The electrical repulsion
between two electrons, for example, is some 10^40 times stronger than
their gravitational attraction.  Nevertheless, gravity is the dominant
force on the large scales of interest in astronomy.  There are two
reasons for this.  First, gravity is a "long range" force---the strong
nuclear interactions, for instance, fall off with distance much faster
than gravity's inverse square law.  Second, gravity is additive.
Planets and stars are very nearly electrically neutral, so the forces
exerted by positive and negative charges tend to cancel out.  As far as
we know, however, there is no such thing as negative mass, and no such
cancellation of gravitational attraction.  (Gravity may sometimes feel
strong, but remember that you have the entire 6x10^24 kg of the Earth
pulling on you.)

For most purposes, Newton's law of gravity is extremely accurate.
Newtonian theory has important limits, though, both observational (small
anomalies in Mercury's orbit, for example) and theoretical
(incompatibility with the special theory of relativity).  These limits
led Einstein to propose a revised theory of gravity, the general theory
of relativity ("GR" for short), which states (roughly) that gravity is a
consequence of the curvature of spacetime.

Einstein's starting point was the principle of equivalence, the
observation that any two objects in the same gravitational field that
start with the same initial velocities will follow exactly the same
path, regardless of their mass and internal composition.  This means
that a theory of gravity is really a theory of paths (strictly
speaking, paths in spacetime), which picks out a "preferred" path
between any two points in space and time.  Such a description sounds
vaguely like geometry, and Einstein proposed that it *was*
geometry---that a body acting under the influence of gravity moves in
the "straightest possible line" in a curved spacetime.

As an analogy, imagine two ships starting at different points on the
equator and sailing due north.  Although the ships do not steer
towards each other, they will find themselves drawn together, as if a
mysterious force were pulling them towards each other, until they
eventually meet at the North Pole.  We know why, of course---the
"straightest possible lines" on the curved surface of the Earth are
great circles, which converge.  According to general relativity,
objects in gravitational fields similarly move in the "straightest
possible lines" (technically, "geodesics") in a curved spacetime,
whose curvature is in turn determined by the presence of mass or
energy.  In John Wheeler's words, "Spacetime tells matter how to move;
matter tells spacetime how to curve."

Despite their very different conceptual starting points, Newtonian
gravity and general relativity give nearly identical predictions.  In
the few cases that they differ measurably, observations support GR.  The
three "classical tests" of GR are anomalies in the orbits of the inner
planets (particularly Mercury), bending of light rays in the Sun's
gravitational field, and the gravitational red shift of spectral lines.
In the past few years, more tests have been added, including the
gravitational time delay of radar and the observed motion of binary
pulsar systems.  Further tests planned for the future include the
construction of gravitational wave observatories (see D.05) and the
planned launch of Gravity Probe B, a satellite that will use sensitive
gyroscopes to search for "frame dragging," a relativistic effect in
which the Earth "drags" the surrounding space along with it as it
rotates.

References:

For introductions to general relativity, try:
  K.S. Thorne, _Black Holes and Time Warps_ (W.W. Norton, 1994)
  R.M. Wald, _Space, Time, and Gravity_ (Univ. of Chicago Press, 1977)
  J.A. Wheeler, _A Journey into Gravity and Spacetime_ (Scientific
     American Library, 1990)

For experimental evidence, see:
  C.M. Will, _Was Einstein Right?_ (Basic Books, 1986)
or, for a more technical source,
  C.M. Will, _Theory and Experiment in Gravitational Physics, revised
     edition (Cambridge Univ. Press, 1993)

You can find out about Gravity Probe B at
<URL:http://einstein.stanford.edu/> and
<URL:http://www.nap.edu/readingroom/books/gpb/>.