How is a promoted pawn designated? Do players just remember that it is no longer a pawn?

In tournament play, the promoted pawn would be replaced with whatever piece it was promoted to (usually a queen). Tournament chess sets typically have a couple of extra queens for this purpose. If you're playing at home and don't have any extra pieces, you could replace it with a piece that's been captured or come up with some way to temporarily mark it so you can tell. Once it's out among the other pieces, it's going to be really difficult to just remember which pawn was promoted (although, if you were recording all the moves, you could figure it out).