If you've got a modern CPU, you need a CPU cooler.
Gone are the days when PC CPUs could get away with just a little aluminium heat sink stuck on top of them, or no cooler at all. When any Intel or AMD CPU on the shelves today is going full blast, it's pumping out heat - less than 20 watts for current Celerons, less than 30 for current P-IIIs, something around the 40 watt mark for current Durons, and better than 70 watts for top-of-the-line Athlons.
You need a pretty big heat sink, with a fan on it, to shift that energy away.
The faster a given CPU runs, and the higher its supply voltage, the more heat it emits. If the cooling thingy attached to the top of the chip can't dissipate that heat well enough, the CPU will get too hot and stop working.
Note that "stop working" does not imply "and will never work again". Pretty much any cooler will stop a CPU from heating up so fast, and so far, that it's destroyed. If you run a current model CPU with no cooler at all, you can blow it up; Athlons and Durons can die in seconds, and often do when someone's failed to notice that their cooler isn't clipped on right and thus doesn't actually touch the CPU.
But if you're just using an inadequate cooler, as opposed to none at all, your computer will merely be flaky. You'll be able to cure the problem by improving your case ventilation and/or upgrading to a better cooler.
But which to choose?
Even if you're running a CPU at stock speed, the cooler you pick can affect your system stability, especially if you're using the hotter AMD chips. If you're an overclocker, high powered cooling is essential.
Just because a cooler looks, well, cool, doesn't mean it's good. Just because it's expensive doesn't mean it's good.
And even if a cooler is efficient at the basic task of shifting thermal energy from a CPU package to the air, that doesn't mean you'll like it. If the thing's got some bizarre attachment system that drives you bananas when you're trying to put it on your processor - or, worse, if it smashes your processor because it clamps down too hard - then I think it's safe to say that it's not a good buy.
So, in the name of consumer information, I got a pile of CPU coolers, and I annoyed the heck out of myself for some days testing them.