We'll start with the most severe cases: as we reported on the system requirements post, Mac G4 are out. G4 Macs are old enough to not usually be used as primary machines any more (you can see if you have one by going up to the Apple and clicking "About This Mac"), so most people won't have to worry. But if you're still running OS X 10.3 on an old G4 Powerbook or Mac Pro, it's probably time to upgrade -- dig deep into your pockets and pick up the new $999 MacBook. It'll run great, and you'll get Leopard, which is terrific anyway. Otherwise, if you're already running one of the newer Core 2 Duo Macs, you won't have much to worry about -- while a new MacBook will help your graphics, anything that runs WoW well now will still run it well a month from now.
If you're running on a PC, Wrath's requirements are a little more harsh than Burning Crusade, but not by much. Usually the best and cheapest thing you can do is up your RAM -- if you're running anything less than 1gb (you can find this out by right-clicking on My Computer and then going to Properties), you might look at buying some more RAM. If you're even slightly computer literate, you can buy it on Newegg and click it into the motherboard yourself, but if not, RAM is the kind of thing you might ask the computer geek in your life to help you out with -- it's easy to do.
Mac users can also upgrade RAM if they like, though odds are it won't help quite as much as PC folks. Apple kindly sells RAM through their retail stores, so you can call up to reserve a spot at the Genius Bar, and then head in and they'll do it for you while you wait.
The next step after that is probably a video card -- you can find some great cards online for a couple of hundred bucks, and especially if your video card is a few years old, getting a new one in your computer will really spruce up your video performance (letting you see all those cool shaders and shadows in Wrath). The drawback here, though, is that video cards aren't as easy to install as RAM -- while they're not hard to actually put into the case (all you need is a screwdriver and some patience), you'll also need to install drivers, and unless you've done it before, that can be kind of tricky. But again, recruiting your local nerd (we really like gift cards to Best Buy and/or adult compensation, if you know what I mean) will usually get the job done.
If your processor is slow -- slower than 1.3 GHz (so something between 700-999 MHz -- again, you can see this by going to the Apple and "About This Mac" or right clicking on My Computer and choosing Properties, and remember you're looking at Processor, since there may be another speed in there), then you'll really run into trouble with Wrath, and you're probably already experiencing slowdowns and graphical problems now. If that's the case, it's probably worth it to upgrade to a newer PC -- when you buy one, look for some with at least 128mb graphical memory (not RAM), and at least a 2GHz processor, dual core if at all possible. That kind of machine (and that includes the new MacBooks and Pros) will run WoW like buttah.
I'm sure our commenters have lots more to say about the subject -- if you're still confused as to how your PC or Mac will run WoW or Wrath after it releases, feel free to drop your system specs in the comments down below, and we'll likely give you an earful on how we think it'll do.