There are lots of variables; the wine type, method of production, age and so on. There are all those considerations and exceptions but for 95% of the wine that people drink every day, the answer is pretty simple.
Three days: At home, I can keep wines up to 3 days after the bottle has been opened, (this doesn't happen very often). Once a bottle of wine is opened, the oxygen in the air starts a process that softens the flavours and opens up the aromas. As this process (oxidation) continues over hours and days, the wine is ultimately undrinkable; the trick is to drink the wine before this point. You can (and usually should) refrigerate or store in a cool place re-closed (open) bottles. You can buy wine-gadgets to create a seal / slight vacuum in the bottle.


You can get systems that put a layer of inert gas in the bottle. All these efforts are aimed at slowing the oxidation that will eventually destroy the wine. What makes the whole thing tricky is that wine will not immediately go from good to bad. Each person has a different point at which they identify the wine as having gone bad.
If you want to play it safe (and who doesn't with wine), use the 3 day rule, (though during summer or in hot climates this can be reduced). Re-close and refrigerate the bottle for up to 3 days. With red wines, remember pull the bottle out from the fridge at least 1 hour before you want to use it so it will warm up to a temperature of around 17-18°C. With white wines or rosés, depending upon the room temperature and time of year, give the wine 10-15 minutes or so to get to about 8-9°C.
If you keep a wine for more than 3 days, you will be serving a wine that has lost most of the personality and characteristics that are prized. The aroma will start to change and much of the fresh fruit smells and tastes will subside. At worst, you'll be serving a wine that has oxidized too much and has gone bad.
Dessert wines, Ports and Sherries can last longer but those are special cases (due to the method of production, increased sugar levels and the higher alcohol content, which can act as a preservative and slow the process). In general play it safe with the 3 day rule, but always smell the wine before pouring any glass of any age.