Let's give some guidelines...
P3 955 MHz processor and 128 MB RAM
Now, I'm not old, 23 in fact, but that computer is *far* from ancient by my standards. Then again, I'm the freak running debian , w/o X mind you, on a 233 MHz Pentium MMX with 64MB of RAM.
So, yes, Linux will allow you to run smooth on your hardware. I've successfully used http://www.damnsmalllinux.org" on said machine with no problems. You might want to consider Ubuntu, considering KDE might give that machine a heart attack (don't get me wrong, I use KDE on my hefty machine), or follow dduardo's suggestion and use fluxbox or XFCE. Fluxbox is the default on DSLinux, btw.
As an aside, you might want to use the word best when describing any GNU/Linux distro with caution. Each distro has strengths and weaknesses, unless you build it from the ground up. If you want to do that, then gain a few years experience looking at all the distributions, tinkering under the hood, screwing installations on the command line, and then grab something like Gentoo and go to town.
But I am biased. I <3 Debian!