Speed of gravity
The speed of gravity is the speed at which changes in the location of an object propagate their gravitational effects to all other objects in the Universe.
Newtonian mechanics
Isaac Newton's mechanical systems included the concept of a force that operated between two objects, gravity. The quantity of force was dependent on the masses of the two objects, with more massive objects exerting more force. This led to a problem: it seemed that each object had to "know" about the other in order to exert the proper amount of force on it. This troubled Newton, who commented that he made no claims to how it could work.
Given two bodies attracting each other, the question then arises as to the speed of propagation of the force itself. Newton demonstrated that unless the force was instantaneous, relative motion would lead to the non-conservation of angular momentum. This he could observe as not being true, in fact the conservation of momentum was one of the observations that led to his theory of gravitation in the first place. He therefore assumed that the speed of gravity was infinite.
Field theories
Michael Faraday's work on electromagnetism in the mid-1800s provided a new framework for understanding forces. In these "field theories" the objects in question do not act on each other, but on space itself. Other objects react to that field, not to the distant object itself. There is no requirement for one object to have any "knowledge" of the other. With this simple change, many of the philosophical problems of Newton's seminal work simply disappeared while the answers stayed the same, and in many cases the answers were easier to calculate.
By viewing gravity as being transmitted by a field rather than a force, it is possible for gravity to be transmitted at a finite speed without running into the problems that Newton sees. If gravity is transmitted by a field, a moving object will cause the field potentials to be non-circular. Hence by using a delayed field rather than a delayed force, one can show that the force will point to where an object is currently rather than were it was in the past. Gravity is still travelling at a finite speed because a suddenly change in the direction of an object will not be noticed by the object it is pulling without a delay.
A similar effect occurs in electromagnetic fields.
General Relativity
In general relativity (GR), the field is elevated to the only real concern. The gravitational field is equated with the curvature of space-time, and propagations (including gravity waves) can be shown, according to this theory, to travel at a single speed, cg.
Measurements of various sorts, notably orbiting neutron stars, have shown that cg must be very close to c, the speed of light.
This finite speed may at first seem to lead to exactly the same sorts of problems that Newton was originally concerned with. Although the calculations are considerably more complicated, one can show that general relativity does not suffer from these problems in the same way that a classical delay potential theory would. However, Tom Van Flandern has made a name for himself by insisting that this proves general relativity incorrect. Be that as it may, most physicists who have interacted with him consider him a crank who is unable to understand that the objection he presents was resolved in the 19th century.
Experimental measurement
In September 2002, Sergei Kopeikin made an indirect experimental measurement of the speed of gravity, using Ed Fomalont's data from a transit of Jupiter across the line-of-sight of a bright radio source. The speed of gravity, presented in January 2003, was found to be somewhere in the range between 0.7 and 1.2 times the speed of light, which is consistent with the theoretical prediction of general relativity that the speed of gravity is exactly the same as the speed of light.
Other physicists have criticised the conclusions drawn from this experiment on the grounds that, as it was structured, the experiment was incapable of finding any results other than agreement with the speed of light.