Once a bottle of wine is opened, the oxygen that works to "open" the wine and unleash the aromas will begin to work against it. This is a problem occurs when you don't want to drink a whole bottle in one sitting.
Refrigeration will preserve many wines for up to 24 hours, but there is no guarantee. If you are saving a red wine for the next day, put it in the refrigerator! It will keep much better than on the counter. Just remove it a couple hours before drinking.
There are some commercial products on the market that try to increase the time you can store your open bottle of wine. Generally, though, you can keep white wine for about three to four days and red wine for two -- no matter what system you use.
Opinion: some have found that these systems do not work that well. Often the best solution is just to finish the bottle or refrigerate. Also consider buying half bottles!