Can someone describe the flavor that beer made partly with rye adds to beer?
You could probably ask 100 different brewers that question and get a hundred different answers. But simply put: it tastes like rye.
Most commercial rye beers have just enough rye in their grain bill so that rye flavor is present, but it's generally very subtle. To me, it's just a slight nutty crispness. There are homebrew recipes online, however, that use rye for more than half their grain bill, and those beers tend to taste very similar to the grain itself.
If you're unfamiliar with the flavor of rye, the best way to get acquainted is to try a small handful of flaked rye or rye malt from a homebrew store. You can also find the same flavor in a loaf of rye bread or a wafer of rye crispbread.