Let it breathe :

If you ever need vinegar (albeit a bit on the expensive side), then leave your bottle of wine open for around 12 hours and oxygen will work its dark deeds on the wine. Breathing however which involves exposing wine to air before drinking it is generally considered to benefit younger wines (those within the first third of their lives - varies from wine to wine) as opposed to older ones (those in the last one/third of their lives).

You can perhaps compare a bottle of wine to a room which has been closed for some time and needs to be aired a bit before it can be lived in, the best way of which is to open the windows. It's always a bit tricky to judge exactly how long the wine should be aerated for. Generally if it's a youngish white wine expert opinion suggests it'll require not more than an hour of aeration whereas a youngish red wine could do with around an hour and a half. It's always better to err on the side of moderation though.

There's another school of thought, which believes that the wine should be tasted immediately after opening to see how long it needs to be aerated for or if it can be drunk immediately. Ok if you're a connoisseur, if you're not then avoid. The question you're probably asking is why the hell do we have to go through this rigmarole of breathing. Well you'll benefit in a couple of ways, one there's a term called "bottle stink" which applies to smells which may or may not creep into the airspace inside. Aeration can help clear this stale air and also secondly help the wine develop it's bouquet.

Cant or decant :

Decanting is an operation, which aims at transferring a wine from a bottle to a carafe in order to eliminate sediments and serve a clear wine. Old vintages, specially red wines tend to precipitate with time. Those precipitations (mostly tartrate crystals and color) affect in no way the taste of the wine, but are not pleasant to the eye if present in the glass at the time of drinking. Decanting the bottle before serving ensures that the wine in the carafe, and hence in the glasses, will be perfectly clear. Prior to the actual act of decanting it's always helpful to keep the bottle upright for a day to let the sediment settle.

A decanter

Practically, decanting is done by pouring slowly the wine from the bottle into a carafe under the light of a candle. Shine the candle or flash a torch from under the neck upwards. When the sediments reach the shoulder of the bottle, one stops pouring. Few milliliters are wasted in the process. Decanting should be done very carefully so that the wine is not too much in contact with the atmosphere.

Old wines are like venerable old persons and should be handled very gently not to destroy any component of their exceptionally complex bouquet. The other purpose which decanting serves is aeration (the need for which has been outlined above).