Many non-technical people I know fail to understand that the single biggest reason why their two-to-four-year-old computers run very slowly is a paucity of system memory (RAM). With every upgrade, i.e. multiple times a year, browsers like Firefox, Safari, and Internet Explorer hog more and more RAM. RAM is cheap and easy to install, even for someone who's not used to opening up their computer.
My favorite web site for buying RAM is Crucial.com. It has a wizard that lets you select your computer manufacturer, then model, and shows you memory upgrades available for your system. For example, I just priced out 2GB of RAM for a MacBook at $34.
How much RAM should you get? The answer's simple. Since it's so cheap now, you should get as much as you can fit in your computer. The Crucial site will tell you how many slots you have, and what will fit in them. If you're running Windows XP, you should have at least 1GB; for MacOS, at least 1GB, and for Vista, 1.5GB.
How do you put RAM into your system? Power down your system first, of course. It's easiest for laptops. On the bottom of your laptop, there's a panel with two phillips-head screws. Open it up, and there are slots for memory. Make sure you seat the chips properly, close up the panel, and you're done. You don't have to do anything to tell the operating system to recognize the new memory, as it will figure it out automatically. See http://crucial.com/install/sodimm.aspx for more details.
For desktops, you'll need to open up your system case. See http://crucial.com/install/dimm.aspx for instructions.
Apologies for the single-mindedness of linking to Crucial all over the place. I have no economic relationship with them - I've used the site a bunch of times, and find it the least confusing and cheapest way to do this upgrade.
In summary, it's a no-brainer. If your system is slow, upgrade your memory! You'll save so much time it'll be worth it many times over.