OK, the assertion that you don't need a new computer needs a bunch of qualifiers. But for years and years, the biggest selling point of new machines has been faster CPUs, or, more recently, more cores. Unless you have specialized needs, your PC's responsiveness during any given task is unlikely to be perceptibly improved by a faster CPU.
If you have ordinary needs -- web browsing, email, word processing -- then about any PC sold in the past 6 years (at least) remains overpowered for your usage.
As a frugalista, environmentalist, and technologist, it breaks my heart to hear of people replacing their computers because they've become intolerably slow over the years. That's not the hardware. That's just what happens to old Windows installations (you won't hear this about this phenomenon from Mac or Linux users.)
If you were to reinstall Windows XP from scratch, your PC would be fast again. To guarantee you don't lose your existing data, my preferred method is to buy a new hard drive, and an external enclosure that works with your existing drive (these are net very expensive.) Remove the existing drive, and install the new one. Install the OS, and update it immediately -- whatever's on your CD is outdated. Finally, put your original drive in the enclosure and plug it in, so you can transfer your old settings and files.
This is a somewhat painful process -- XP's installer wasn't really designed for end-users.
I'm talking about XP and not Vista or Windows 7 for two reasons -- I've never used them, and their stated requirements call for more recent hardware (though the stated minimums would still be exceeded by anything sold in about 6 years, though you might need more memory.)
Next up, I'll describe some of the hardware changes that actually do make a difference.