Wines stay fresh for a couple of days after opening; about 2 days for white wines and 3 days for reds according to Real Simple. Because whites have more acid, they tend to stay fresh slightly longer.
The easiest way to keep wines fresh without any special equipment is to dip the cork in a bit of wine, and then push it back into the bottle. However, if a significant portion of the wine is gone, a smaller bottle will keep more air out of the wine, which will help it stay fresher. For longer periods of time, up to 10 days, a vacuum wine pump will remove air to keep the wine from oxidizing as quickly. Keeping wine for too long will change the flavor significantly.