A constantly dripping showerhead doesn't just waste water - it can drive you crazy.
Unscrew the showerhead from the pipe coming out of the wall. This can be done by hand but sometimes requires a crescent wrench or large pliers. The head may be held on with a screw, which you'll have to remove.
Look at the threads inside the showerhead where it screws into the pipe. There should be a small washer made of plastic or a rubber O-ring. Replace it if it looks even a little damaged or brittle.
Wrap the showerhead stem with Teflon tape or pipe sealer (pipe dope) to seal the connection.
Remount the showerhead on the stem. Don't overtighten. Hand-tightening should suffice.
Turn the water on and off. Wait several minutes and check for drips or leaks. If the showerhead is still leaking, you may have problems with the shower's water-control valve and need to call an expert.
When working with chrome or brass fixtures, protect the surfaces from tools with a piece of leather, heavy cloth or duct tape.