[edit] Steps

Unscrew the showerhead. Use a wrench if needed.
Remove the rubber gasket inside the shower head to inspect. If the gasket leaves black goo on your fingers, it needs to be replaced.
Match the old ring to the new to select the right size.
Place the ring into the shower head. It will take a small effort to push into place. Make sure the ring lays flat inside the head assembly.
Wrap the Teflon tape around the pipe threads of the pipe (not the shower head). Wrap in the same directions the threads flow, and completely cover the threaded area without covering the smooth pipe. Use about two wraps of tape pulled fairly tight. You should be able to see the ridges of the threads but they shouldn't be cutting through the tape.
Tear tape to separate from spool.
Thread shower head into place, turning until the head is hand-tight.
Test repair by turning the water on. If there is no leak from the seam, you are finished!
If there is still a leak,unscrew head and re-screw.
Test.
If leak is almost gone, using the wrench, turn the shower 1/2 turn further. Do not overtighten, you risk stripping the threads.
Test.
If leak persists above an almost imperceptible "welling", start over, replacing the Teflon tape a little tighter, and with an extra wrap of tape. If the pipe is very old, the threads may be worn enough to require the extra tape.

[edit] Tips

If your shower head won't loosen easily, coat the joint with marine jelly, or white lithium grease and let it sit for an hour or more. Try not to force the head! You may strip the joint, crack the head, or (worst of all) crack the feeder pipe.
Another option if the shower head won't loosen is to replace the entire shower arm. Typically a new 6" shower arm can be purchased at your local home improvement store. Follow the instructions on the package to unscrew the old shower arm and then screw in the new shower arm. Be sure to use plumbers tape or pipe dope to the end screwed into the main pipe. Then attach the new shower head to the new arm. Be sure to check for leaks.
If you don't want to replace your shower head, look for a leak repair product. The quickest solution is light-activated resin, fixes leaks in 30 seconds.
Make sure to use true Teflon tape and not just any plumbers tape. True Teflon tape works much better and it is worth the little extra it cost.

It is important not to overtighten the joint, as you risk stripping the threads and cracking the head assembly.
Start with no more than 2 wraps of tape. Using too much tape results in a bad fit and your first test may wash the ceiling!

Teflon tape. This is available in the plumbing section of most hardware departments.
Assortment of O rings for faucet repair.
Pipe wrench or crescent wrench.