Steam synchronizes many backup files on its servers. They are automatically downloaded via Steam when you install a game, but this is not the only way to get them. You can also download them directly from the Valve website in your browser.
Enable Steam Cloud Sync in Steam
If Steam does not automatically download your old backups after installing a game, make sure Steam Cloud is enabled for that game in Steam.
To do this, locate the game in your Steam library, right-click on it, and select "Properties". Click the "Updates" tab and verify that the "Enable Steam Cloud Sync" option is checked. If this option is not checked, Steam will not automatically download your backups to the cloud, nor download new ones.
If you do not see a Steam Cloud option for a game here, this game does not support Steam Cloud. All games on Steam are not – it depends on each game developer.
Download the files in your web browser
Valve also allows you to upload your backup files to the Steam cloud via a web browser. You can only download your backup files without downloading the entire game again.
To find your backup files, visit the Valve website. See the Steam Cloud page in your web browser and log in with your Steam account.
You will see a list of games using your Steam Cloud storage. Locate the game in the list (press Ctrl + F to use the browser search) and click "Show Files" to view all the files in a game.
Each game has a page listing all the files stored in Steam Cloud, as well as the date they were modified. To download a file, click on "Download". Download all the backup files for your game. You will get a copy of its backup games.
This is especially useful for multi-platform games that do not automatically synchronize backup files between platforms.
For example, Borderlands 2 is available for PC and Mac. It synchronizes its backup files with Steam Cloud, but PC and Mac versions synchronize files separately. You can download Mac (or PC) versions from the Steam website, and then manually import them into the appropriate folder to move your Borderlands 2 backups between platforms.
Without using the website, you will need to download Borderlands 2 on one platform, extract your backups, and then download it on a second platform!
This is a known problem for other games as well. For example, it also seems necessary to move backup sets from Torchlight for Windows to Torchlight for Linux.