In the United States, Rye Whiskey has a few legal requirements that are very similar to bourbon:
It must be made from at least 51% Rye. (The remainder is usually corn and malted barley. Bourbon must be 51% corn.)
It is distilled to no more than 160 (U.S.) proof, and aged in charred, new oak barrels. The whiskey must be put into such barrels at not more than 125 (U.S.) proof. (Again, similar to bourbon.)
Rye that has been so aged for at least 2 years may be further designated as "straight", as in "straight rye whiskey". (Seeing a pattern, here?)
Rye is (generally) stronger, drier (and some would argue, smoother) than bourbon. It was very prevalent in the Northeastern states but began to be replaced with bourbon after Prohibition. A few brands, such as Old Overholt, survived Prohibition, and both Jim Beam and Wild Turkey produce their versions of rye.
Although the continued production has largely been due to bourbon producers, the home of George Washington, Mount Vernon, has recently begun to distill and sell a version of the rye Washington distilled. Rye is currently undergoing a small but growing revival in the United States.
Canadian Whiskey used to be predominantly rye, but this is no longer the case, though most will contain a small percentage of it.
Rye and bourbon can be used fairly interchangeably, but the addition of rye will result in a lighter, drier cocktail.