Can anybody tell me-if I actually have some red wine left in a bottle, should I leave it on the counter with the cork in it or do I have to refrigerate it?

And also--should white wine be refrigerated after opening? I usually chill it, then let it sit out for about 15 minutes, and then serve. Sometimes, I have a bit left--do I refrigerate it again?

None of my wine books mentioned this, so I guess it's one of those things a person is supposed to be born knowing! Like how to change a tire . . .