Can't run natively, usually will compile.
There are a lot of technical reasons why running Linux programs on OS X won't work. It would be like running Solaris programs on Sun OS or Solaris programs on Linux; it just won't work.
In order to run Linux programs, you need to recompile them. For command line programs (ie text based programs that do not run in their own windows), it *generally* only requires a recompile. Any program that uses only standard Unix libraries will probably compile; but differences in the OS's may require some minor code changes. If you download the source for a program and follow the build instructions in the distribution, you'll see wether or not it works.
For GUI programs, you'll need to get an X-Windows environment running first. This means you need an X-Windows server and a window manager. This is a comewhat complicated process.
In both senarios, wether or not a program will run will depend on wether the needed libraries are available on your system. If not, you need to compile and install the necessary components.
But there is an easier way to do all of this. Get Fink. Go to fink.sourceforge.net and get it. Its basically an automatic download and install utility for open source programs on Unix platforms. It has all of the programs pre-compiled to run under OS X and will install them for you. You can even use this to install X-Windows on OS X so you can run those GUI programs and OS X programs simultaneously.
There is a lot to learn, but there is a lot that can be done with these programs. Good luck with it all. And people here are very nice about answering your questions. Ask if you get stuck!
Matthew