Update: Apr 2015
After further testing with a bunch of distros on live CDs and by installing them on some Pentium 4 era test computers I have found I like LXLE and Mint XFCE the best. Both of them have lightweight desktop environments. LXLE seems faster and has had the best hardware detection on the test computers. It hasn't choked on SiS video cards or old Broadcom wifi cards, like the others, and it looks really nice and runs very fast.
Recently Microsoft stopped updating Windows XP. This means no more security patches, and no more bug fixes.
Without security patches and bug fixes your Windows XP computer will become increasingly obsolete, and increasingly vulnerable to viruses and web page based hacker attacks. It also means that the whole ecosystem of programmers who program for Windows are going to stop updating their apps for Windows XP.
So, you need a new operating system. Notice I didn't say, "A new computer."
While you may feel that you'd rather just get a newer (not New, necessarily) computer, you don't have to. You can update your old Windows XP computer with a new operating system that will still be maintained. Doing this could allow you to keep using the same computer instead of paying hundreds for a new one.
The computers I have been testing all have at least 512 MB of RAM and are all Pentium 4s.
For really old computers (Pentium III, less than 512MB of RAM), here are what seem to be the best two choices for new operating systems;