Using backups of games

Did you already uninstall Steam?

Reinstall Steam and then copy your backed up game folder back to the Steam game folder and it should detect the games being installed on restart.. if it doesn't it should detect them when you try to install them.

The process can be tricky, as sometimes the files don't detect and it will only download them from scratch.
I found this out the hard way after trying to manually keep the /common/ files as they were.

The easiest way is to use the in-steam backup function, where you can create a backed-up copy of a steam game specifically designed as a restorable copy, which you can then transfer onto a different HD for safe-keeping. For instance, I have an external HD where I keep backups of particularly large steam-games, to save me from having to download them again (given my ISP has a monthly data limit and doesn't have the fastest of transfer speeds). Restoring from a backup is as easy as taking the backup sub-folder it generates for you, moving it into /steam/backups/, and then running the restore function from within the steam client.