Beer finnings are used for two separate things at two different stages. There are kettle finnings which include Irish moss, whirlflock, and eisenglass. These are used in the last few minutes of the boil to attract the proteins in the cold break together so you have a large goopy mess in the bottom of the kettle after you use the wort chiller. This is supposed to prevent the chill haze causing proteins from getting into the fermenter. They usually work best with the whirlpool method of cooling.
Kegging finnings include gelatin which alleviates the need to filter a beer. It works best when the beer is crashed down to just over freezing and the gelatin tabs are added. The gelatin will create a jelly of yeast that collects on the bottom of the fermenter making it possible to rack the crystal clear beer right off the yeast cake.
Wine finnings from what I understand are designed to keep the proteins and tannins from entering the bottles after fermentation.
