It depends on the wine. Younger wines keep well longer than older naturally, and white wines (as well as lighter bodied, higher acid reds) keep longer too mainly because they have lower pH and higher T.A. For instance, a bottle of anything good more than 10 years old would probably deteriorate considerably after 24 hours without any intervention. But most wines will keep 3 days in the fridge, and some high-acid whites up to 5-6 days.
However, really tannic wines can also resist oxidation quite well. As a few posters have alluded to, some get better with time - these tend to be the ones which probably benefit most from being cellared anyway.
Another thing to consider is the amount of ullage you leave in the bottle. Obviously if you only had one glass, the remaining four glasses will resist oxidation much better than if you had four glasses and left only one in the bottle. Bottles with single glasses left in them rarely keep for more than one day.
There is no problem with storing reds in the fridge, as the lower temperature also retards the oxidation reaction. The only thing is you have to wait for the glass of wine to warm back up to a drinkable temperature. If you're not intending to drink the remainder of the bottle, just pour yourself the glass and let it warm up, rather than letting the whole bottle warm up.
Some replies have also mentioned ways to preserve your wine. The vacuum suction stoppers are popular and fine if you're not dealing with very fine wine. Otherwise, it can rob the bouquet of high-notes since it creates a strong evaporation potential within the bottle for the delicate, volatile aromatic compounds. If you find yourself with open bottles of wine very often, seek out an inert gas preservation system. This is what we use at the wineries, and can be a combination of nitrogen and argon, or the more professional systems just use argon as a blanket. Stay away from Carbon Dioxide preservation systems for still wines, as they can change the T.A of the wine over time.
If you do happen to have Champagne or Champagne-style sparkling wines open and left over, use a bouchon. It can help somewhat, although usually by the time you realise you've got left overs, you've probably already lost most of the pressure from the wine. If however, you know you're only going to have one or two glasses. Pour those and place the bouchon on immediately and you've just bought yourself 3-4 days to finish the rest of the bottle. :-) Non-Champagne style sparkling wines, like Asti and cuve close styles don't have that much pressure to start off with and will lose their sparkle if you try to store open bottles, so just finish them off.
I'm assuming that you're talking about wine which is at least worth a keeping for a second drink, and not the stuff which was probably made to be cooked with in the first place!