If you search the Internet or the Steam discussion forums, you can also find additional suggestions about how to install the client on other distributions, including Debian, Fedora, and Linux Mint. Similarly, you may need to search for how to install the 32bit client on a 64bit system. Be sure, too, to have your distro’s latest release.
If you are merely curious, better that you stick with Ubuntu, which has provided detailed instructions. But even that is not enough. You need a video driver with hardware acceleration, not just for most of the games, but –strangely — simply to run the client. You also need the latest version of Flash from Adobe to view content on the Steam site — an earlier version or Gnash won’t do, although nothing warns you of the fact.