General Relativity has a set of equations that give results for how a
lump of mass-energy changes the space-time around it.  (See D.03.)  One
of the solutions to these equations is the infamous black hole, another
solution is the results used in modern cosmology, and the third common
solution is one that leads to gravitational waves.

Over a hundred years ago Maxwell realized that a solution to the
equations governing electricity and magnetism would create waves.
These waves move at the same speed that light does, and, hence, he
realized that light is an electro-magnetic wave.  In general,
electromagnetic waves are created whenever a charge is accelerated,
that is, whenever its velocity changes.  

Gravitational waves are analogous.  However, instead of being
disturbances in electric and magnetic fields, they are disturbances in
spacetime.  As such, they affect things like the distance between two
points or the amount of time perceived to pass by an observer.
Moreover, since there is no "negative mass," and momentum is
conserved, any acceleration of mass is balanced by an equal and
opposite change of momentum of some other mass.  This implies that the
lowest order gravitational wave is quadrupole, and gravitational waves
are produced when an acceleration changes.

Because gravitational waves are waves, they should exhibit many other
properties of waves.  For example, gravitational waves can, in
principle, be scattered or exhibit a redshift.  (But see the next
question on the difficulty of testing this prediction.)

[Note, *gravitational* waves...gravity waves are something else
entirely (they occur in a medium when gravity is the restoring force)
and are commonly seen in the atmosphere and oceans.]