First things first: While you can run the new Linux Steam gaming client on other versions of Linux besides Ubuntu, for now it's only supported on Ubuntu.

If you want to try Steam on other Linux distributions, start at Valve's Steam under Linux page. Good luck!

Moving on, to run Steam games you should be running Ubuntu 12.04 or newer on a dual-core system with 2GBs or more of RAM running at least 2.8GHz. You'll also need an nVidia GeForce 8500/9600GT or an ATI/AMD Radeon HD 2500/3600 or better graphics and their latest drivers. Finally, you'll need the newest version of Flash, that's 11.2 for Linux, and OpenGL 2.1.

In addition, if you're running a 64-bit Linux distribution, you may need to download some 32-bit libraries during the installation process. At this time, there are no fully native 64-bit version of the Linux Steam client.

That said, if you've met all the preconditions, open the Ubuntu Software Center and select Steam.