Fining wine consists of adding a 'fining agent' to the wine and mixing them together. Fining agents are typically naturally occurring proteins or substances that have been synthesised to mimic the action of proteins. Phenolic substances have a strong natural chemical affinity for proteins. So when they come in contact with each other, they react, and precipitate out of the wine. So in effect, protein fining agents are used to strip out the phenolics from wine.
Fining can be done for several reasons including: adjusting colour, odour, flavour, tannin, astringency, stability and remove microscopic particles that could cloud the wine. A number of different fining agents are used depending on the type of wine to which they are being applied.

 

Examples include: bentonite, gelatine, egg white, milk casein, isinglass (fish bladders), seaweed, clay and others.
Usually bentonite and egg whites are used by themselves, but gelatine is commonly used to fine white wines. These fining agents have a specific gravity that is slightly greater than the wine. As the substances sink through the wine it binds with any remaining unwanted particles that are suspended in the wine and carries them to the bottom, leaving the wine free of cloudiness.
White wines are fined to remove particles that may cause the wine to brown or lose colour as well as removing heat-unstable proteins that could cause the wine to appear hazy should it be exposed to high temperatures after bottling. Red wines are fined for the same reason but also for the added benefit of reducing the amount of bitter, astringent tannins, making wines smoother and more approachable sooner after bottling.
New Zealand has wine labelling laws that require the use of fining agents that may be an allergenic to appear on the label, as there may be trace amounts left in the wine.
As with filtration, there is the risk of some loss of flavour with fining due to desirable flavour molecules being precipitated out, though, fining is considered less harsh than filtration.