Saving specific games

Is it possible to save certain games to a certain folder, for the purposes of organizing specific games that one might want to study?

Not on the site. You could insert the games into a blog or something but that is about it. 

Is it possible, though, to save a specific game to a chess program, such as Fritz?

You can save the pgn in a folder. Should work perfectly.

Of course. I haven't played chess in a while but now that I'm back, I remember that acronym, pgn. Thanks, skotheim2.

I play on the mobile app and I downloaded droidfish from my app store. You can go in your game archive select a game then download the pgn, you will get a email and when you open it, it loads the game directly on the engine app and you can save it there and analyze all your games. Its pretty sweet.

Sorry, I misunderstood exactly what you were asking.  As others have posted,  it is easy to grab your games from the site and they can be managed on your local system. 

For people in 2024 searching this (like me), you can save the games in the library on chess.com and sort them in different lists.