General relativity, or GR, is Einstein’s general theory of relativity (or the general theory of relativity, gtr), which is the single work his status as outstanding genius rests on (of course, there are plenty of other works which make Einstein a remarkable physicist, but GR stands head and shoulders above the rest).
It is called ‘general’ because it is intended to apply generally, to everything in the universe, at all times, and in all circumstances (compared with the special theory of relativity – or special relativity (SR) – which applies only when and where gravity is not important).
General relativity rests on an astonishingly simple assumption (or postulate), which can be written as “everybody, everywhere in the universe and at every time, will find the laws of nature to be exactly the same“. Well, actually there’s a bit more, which John Wheeler summed up best: “spacetime tells matter how to move; matter tells spacetime how to curve.”
As a theory of gravity, general relativity takes over from Newton’s universal law of gravitation … not that Newton was necessarily wrong, but that GR provides a more accurate way to describe gravity (in the sense that observations and experiments are consistent with GR, but not – in their entirety – with Newton’s law). Because general relativity goes beyond Newtonian gravitation, and because Newton’s theory works very well (you don’t need GR to navigate spaceprobes around the solar system, for example), testing GR is not easy. Nevertheless, over the nine decades since GR was published, thousands of such tests have been done … and general relativity passed every one! Clifford Will keeps a thorough list of all of these, in the Living Reviews of Relativity The Confrontation Between General Relativity and Experiment.
Further reading: General Relativity (University of Illinois); for an introduction to the basic equation John Baez’ The General Relativity Tutorial; and for GR and cosmology, Ned Wright’s Relativity Tutorial (UCLA) should get you started. There’s lots more on the internet of course, but do be careful (sadly, there’s lots of cranky and crackpot material besides the good).
Given the universal scope of general relativity, and given that GR is the basis of modern cosmology (and much more besides), it is the central topic of a great many Universe Today stories; here are a few: Cassini Confirms General Relativity, Flyby Anomalies Explained?, and LISA Will Watch Snacking Black Holes.
Einstein’s Theory of General Relativity, an Astronomy Cast episode, covers this topic well.