Most beer styles use just barley for their grain bill, but other grains like oats, wheat, and rye have a long history in brewing. Rye is the grain of hardier Northern climates, where other grains are difficult to grow. It is strongly flavorful and usually imparts a spicy, bready note when used for beer. Growing interest in rye as a brewing grain means that it is getting used in more and more styles.
Roggenbier - This is a historical German style of beer that has been revived only recently. Roggenbier is brewed with very high proportions of rye, 50% or more, and is similar to a Bavarian Hefeweizen or Dunkelweizen in many respects. Hops are minimal but the rye flavor is huge, and banana/clove character from a traditional Hefeweizen yeast is common. Roggenbier is a challenge to brew because of the high percentage of rye malt, which gets very sticky and gummy in the mash.
RyePA - American rye beers range quite a bit, from easy-drinking rye ales that are similar to a wheat beer in recipe, to massive high-alcohol concoctions. But the RyePA is probably the most popular in the US. This is an American-style IPA with rye included in the grain bill, which lends a spicy, earthy note. A good RyePA will use the flavor of the hops and the flavor of the rye as counterparts, with neither one dominating the beer.
Sahti - An unusual Finnish style of beer made with malted barley and rye and flavored with juniper berries. The traditional method uses entire juniper boughs to filter the mash, which provides the characteristic juniper taste along with some astringency from the wood. There is also traditionally no boil for sahti, the hot mash is simply run off, cooled, and fermented, and this gives the beer some complexity and sour notes. It also makes it quick to spoil, so many craft and homebrewers making sahti decide to boil it. The end result is usually cloudy, with banana and clove notes from the yeast, and noticeable juniper flavor.