Fining is the process of clarifying and stabilizing a wine. A fining agent is mixed in to bind with particles suspended in wine that would make it appear cloudy when poured into a glass. Since fining agents are slightly heavier than wine, the bound particles precipitate out and settle to the bottom of the tank.
Sometimes you might fine a wine to clean it up if it is cloudy, other times you might wish to soften the wine by precipitating out compounds that are astringent. Most red wines won't need fining but whites benefit from the process so they remain crystal clear as they age in the bottle.