The short answer to this question is "not much at all." In fact, I don't think you could buy a computer at your local Best Buy or online that can't run Linux.
Let's take Ubuntu 9.04, which is due to arrive this week. The official minimum requirements for this popular Linux distribution are a 700MHz processor and 256MBs of RAM. I think that's too low. In my experience, you could run GNOME 2.26 on that slow a processor, but the RAM's way too low. You could pull it off with 512MBs, but you'd be happier with a Gigabyte.
A reader recently claimed to me that he had an older PC that can run Windows 7, but wasn't powerful enough to run any modern desktop Linux. Yeah. Right. Windows 7 'officially' requires, a 1GHz processor, 1GB of memory, 16GBs of free hard drive space and 128MB of graphics memory on a chip set or card able to support DirectX 9. As with every version of Windows going back to 1.02, Microsoft is lying about the minimum requirements.
In my Windows 7 testing, starting with Build 7000 to Build 7077, I've found that 7 requires at least a 1.6GHz processor and 2GBs of RAM to run at an acceptable level. Still, that's not bad, and it's certainly better than Vista. That said, there's no way any desktop that can run Windows 7 can't run Linux. Period. End of statement.
What about older PCs though? Linux works great on them. The oldest working server I have is one of the ones I used in 1999 to prove, for the first time, that Linux was a better file server than Windows NT. It's a white-box with a 266MHz Pentium II and 64MBs of memory. These days it's running openSUSE 10.3.
My least-powerful Linux desktop I'm currently using is HP Pavilion 7855 PC. It was born in 2001 with a 1GHz Pentium III and 512MBs of RAM. These days I run Mint 6 on this old vet of a PC. Frankly, that's a little too much operating system for it.
I'd be better off on this level of PC running a Linux that uses a lightweight desktop manager like Fluxbox or XFCE, instead of GNOME or KDE. There are also many desktop Linux distributions that are designed to work this kind of old hardware.
DSL, which comes in a 50MB package, isn't a one trick pony. It's a real, live desktop Linux distribution with the Fluxbox windows manager, Firefox for Web browsing, office programs, IM client, and all the other trimmings. It's my first choice for really old PCs.
In short, if you have a PC that's still not old enough to vote, you can easily find a well-regarded and supported desktop Linux that will run great on it. I wonder how many of you are already running Linux on PC 'antiques.' Drop me a note on this article's comments areas and let me know just how slow you've been able to go with desktop Linux.