Red wines oxidize differently from whites or rose. The polyphenolic compounds are the grapes’ first line of defense against oxidation. The polyphenols translate into tannic structure in reds. So then, red wines with higher tannins are going to oxidize slower and, therefore, taste better the second day than wines with less tannic structure. It’s as if the oxygen comes at the wine and the tannins say, “No! Take me first!” Red wines without this line of defense are going to suffer from a lack of fruit structure and a sense of higher acidity (although the acid structure is the same, it just will seem more pronounced due to oxidized tannins and fruit.).
What do I do? I drink the whole bottle.
What would I do if I were you? Well pumping some of the air out (using a vacu-vin stopper or similar) and then refrigerating will slow oxidation. The compressed nitrous works well too, but is far more costly.
Another idea I’ve just come across, Buy a bag of clear glass decorator marbles, like you would find in the bottom of a flower arrangement. After you’ve poured your one (drink more, already!) glass you displace the empty space in the bottle with enough (clean!) marbles to fill that space. Recork, and you’re good to go. The marbles will sink to the bottom of the bottle, so be careful when pouring the remaining glasses. There will be a bit of a mess cleaning the marbles when the bottle is finished, however.
To all posters here: That bottle is no good after two days. Enjoy drinking it by yourself (it will get the job done) but please don’t serve it to guests.