The most obvious reason to decant is that the wine has thrown a deposit, and that's really only likely with vintage or crusted ports and aged unfiltered reds. For the process to work effectively the bottle needs to have been upright for several hours, then be carefully poured into the decanter in front of a light (traditionally a candle) so you can see as the sediment inches towards the neck. You need to do this in a single movement so that it doesn't fall back and get mixed up with the wine again.
You may also want to get some air into a wine that smells slightly stinky - there are more of these around given the growing popularity of natural wines made with no or very little sulphur - or full-bodied young reds that need a bit of aeration to mellow over-aggressive tannins. You might even want to decant a full-bodied white if you feel it's tasting a bit funky or not showing at its best. When deprived of oxygen wines can develop strange aromas. If they get worse after decanting the wine is probably corked.
How far ahead to decant depends on the wine. it may simply be a question of getting a bit of air into it, in which case simply decanting before you pour will do the trick (much more effective than pulling the cork and leaving the bottle open for an hour or so).
Wines you don't want to decant too far ahead - or at all - are fragile old reds which might fall apart once exposed to air, most ports (you don't need to decant a late bottled vintage or a tawny, for example) or crisp dry fino or manzanilla sherries, which should be served chilled from the fridge - however much that may horrify elderly relatives.
Decanters don't need to be expensive cut glass crystal heirlooms. You can even use a jug. If you're embarrassed by the whole process you can simply double decant: pour the wine into a jug and from the jug back into the bottle (rinsed if the wine has thrown a deposit).
The ideal shape, if you're looking for presents, is one with a wide base like a Ships Decanter or a Wine Carafe which gives you plenty of room to swirl the wine - gently - around. And which actually looks quite stylish on the table.