What actually is Garry's Mod? Is there a game in there?

It's a physics sandbox on the Source engine. Out of the box there's no game really, you spawn props and mess around. The fun is in the gamemodes the community made - Trouble in Terrorist Town, Prop Hunt, Murder, DarkRP. One tip: a lot of servers use Counter-Strike: Source content, and without it you'll see pink checkered textures and big red ERROR signs everywhere.