How do I move Steam games from one computer to another without downloading them again? [duplicate]

Assuming your brother is still living in the same house, you can easily transport them directly over your local area network. I don't know what version of Windows (or other OS) you're using, but it should be pretty doable with a Google search of "(your OS name here) local network file transfer." This utilizes the local connection between the computers in your house to transfer files, cutting out the Internet as a middleman. If you don't have a LAN set up at your house, your cheapest option is a regular Ethernet crossover cable run between the two of your desktops.

Once you've got the files in the place that you want them, you can go into Steam and add the folder you've copied your games to as an install location on Steam so that they'll show up in your library. By default, Steam doesn't look for any games on your computer outside of its install directory, but you can easily add new locations for it to search for games. This guide explains it fairly well.

You can also create a new location for Steam to download games to when you want to download a new game, if you want, or even use the same place that you copied your games to. In fact, I recommend this. I don't know how finicky Steam's metadata detection is (read: how well it knows whether or not the game folder you copied is registered with Steam) but your safest bet is to copy all of your game folders exactly as they were on your old PC into a new folder on your new PC. For example, let's say I installed Steam in C:\Program Files\Steam. I would go to C:\Program Files\Steam\steamapps, which is where the actual game content is stored. I would look in the subfolders (some of your games may be in either your username folder or "common" -- check both) and copy those game installations exactly as they are into a pre-existing games folder on my new PC; for instance, C:\Steam Games\. Then I would add C:\Steam Games\ into my library folders using the guide I posted in the last paragraph. I actually put my Steam games on an external hard drive, and I can freely plug and play the drive and play all of my Steam games no problem.

Hope this helps!