In the early days of Android, gaining root access was a complicated process, involving around a dozen steps and patience enough to test even the most serene of ascetic monks. Nowadays, not only do you not need a saffron-colored robe, it's much simpler: There's an app for that.
However, do note that rooting your phone voids your warranty, and the process is risky even in this streamlined form: there is still a chance that you can brick your phone and render it permanently inoperable.
That being said, also note that these are general instructions. There might be device-specific idiosyncrasies you ought to know about before you begin.
Now connect your phone to computer and make sure that the SD card has not been mounted. You can double-check this by going to Settings, SD card, and phone storage, and Unmount the SD card. Enable USB debugging by going to Settings, Application Settings, Development and Enable USB debugging.
Go back to your computer and double-click on SuperOneClick.exe. Choose the Samsung Capitvate tab if that's your phone, for all others choose the Universal tab unless your phone is built by HTC. HTC phones can only be rooted with the Unrevoked tool, which is designed to break through the NAND-flash memory lock that the manufacturer has installed.
Last, click the Root button and go get yourself a tasty beverage because the rooting process can take a while. If you want to sit and watch it, though, you ought to see a bunch of activity scrolling by, with "OK" appearing after each one. You might see some warnings, too, but they're nothing to worry about as long as SuperOneClick doesn't freeze. In other words, if you're using a laptop, make sure it's plugged in. Once finished, reboot the phone.
Rooting ought to add an app to your phone called Superuser. It's got a skull-and-crossbones icon. It's also a good idea to allow non-Market apps to install if you haven't set that up yet. Go to Settings, Applications, and check Unknown Sources.
Voila! You've now given yourself root access, which ought to allow you to remove apps that come pre-installed, overclock your phone, change the kernels that power it, and, of course, install a customized version of Android itself.