A Trophy is an item of interest taken from the corpse of an enemy. Most trophies do not have much of a purpose other than being sold to a vendor for a little money. However, some less common trophies (usually of the rare quality, indicated by a purple name) are valued by crafters for use in various recipes and as critical success items. Some trophies are house decorations, that often show others that you beat the monster it came from. (celebration sign... accomplishment.. etc. there are many reasons why it's there.) The are rare however, and you might need to beat the boss more than once. An example of a common trophy to sell to vendors is Dry Crawler Barb, while a list of all useful crafting trophies can be found here.