There are a number of linux distributions out there to try. All linux is not the same, so I urge you to try a few different ones before you give up on it or settle with the first one that installs correctly.
The first thing I'd do if I were you would be to go to the following site:
. They have a list of the most popular (by virtue of number of clicks) distributions on the right hand side.
Many linuxes have what is called a "Livecd" - it is a way of booting your pc from a cd to try out a version of linux before you put it on your hard drive. Most of these LiveCD's have a desktop icon that you can click on to install the operating system on to your hard drive. The graphical installers have come a long way since the early years of linux, and if you intend to install only linux on your old machine, you can pretty much install any of them and just take defaults for all the installer questions.
The following are my recommendations:
PCLinuxOS - they have a livecd that is pretty good. Installing is easy, and it has a lot of software out of the box:
Ubuntu - this is a very popular distribution. I'd have to say it probably has the most desktop users at the moment. It's not my personal preference, but there is a vast community of users who swear by it:
Honorable mention: Mandriva, openSuSE