Just as PCs seem to (or sometimes really do) get slower as they age, PCs also tend to get louder as they age. Considering many of them are plenty loud when new, that’s not good.
When a PC is loud, it’s due to one of two types of components: hard drives or fans. The key is to isolate the noise. To do that, your best bet is to open the case, then power the computer on. Running your PC long-term with the cover off isn’t exactly good for your computer, but running it that way for a few minutes won’t hurt anything.

But before you turn the PC on, blow the dust out of it with a can of compressed air. Resist the temptation to just use a small vacuum cleaner attachment; those are static magnets. In some rare instances, just blowing the accumulated dust out will quiet the PC. In nearly all instances, it will make the PC run cooler, and it’ll make you feel better to not have all that crud accumulated inside your expensive equipment.

Loud buzzes are usually caused by failing fans; clunky noises are usually caused by a loud (and often not long for this world) hard drive.

Oiling a fan will usually quiet it and dramatically increase its life expectancy. As long as the fan hasn’t completely died, this is a good bet. It’s certainly cheaper than replacing a fan, and sometimes it’s easier. Don’t ever try to replace the fan in a power supply–oil it, very carefully, and if the noise doesn’t go away, replace the power supply. There are voltages inside power supplies that will throw you across the room, if they’re in a good mood. If they’re in a bad mood, they can potentially kill you, and I really don’t want that.

You can test a fan by stopping it with a pencil or a similarly shaped object. If the noise goes away, you’ve found your culprit.

There’s not much you can do if the hard drive is loud. I’ve heard of people taking hard drives apart and oiling them in efforts to quiet them. Don’t do that. You might well quiet the drive. You also will certainly prevent it from ever working again.

Instead, replace the drive. Modern drives run pretty quietly. Most retail-boxed drives come with free software to copy your old hard drive to the new one, making the upgrade painless. If you’ve never done this before, buy a drive at retail–an OEM drive from a clone shop or mail order outlet may be a little bit cheaper but won’t have any instructions or software–and set aside a Saturday afternoon. Even if you’ve never undertaken anything like this before, it generally doesn’t take more than a couple of hours. As of this writing, a 20 GB Maxtor drive costs $69.99 at CompUSA. The OEM version of the same drive costs $68 at Newegg.com. As you get into bigger drives, the price gap tends to increase, but for many people, 20 gigs is plenty.

Once you’ve oiled or replaced the fans and/or replaced the hard drive with a newer, faster, and quieter model, your formerly loud PC ought to run pretty quietly.