A leaking or dripping faucet (or tap) is generally a sign that a part is worn and needs to be replaced--a fairly simple task. The exact repair depends upon which part of the faucet is dripping and, more importantly, the mechanical makeup of the faucet.

There are four main types of faucets, based on their internal mechanisms: compression, disc, cartridge, and ball. To understand the differences between these, please see the article Kitchen Faucet Types (the same basic principles hold true for bathroom faucets). Your first step in fixing a faucet is to determine what type it is. 

If it has a single handle that controls both hot and cold, it is either a ball, disc, or cartridge faucet. If it has two controls, one each for hot and cold, it is probably a compression faucet, though some such controls have disc mechanisms.  The best way to tell is to dissassemble it.  See How to Disassemble a Faucet.

Refer to the illustrations for the locations of washers, seals, and O-rings--drips and leaks nearly always mean these parts need replacement. Failed washers or seals usually cause drips--to replace them, you must disassemble the faucet. Leaks around the handle usually mean that the O-rings on the stem need to be replaced or that the packing nut or adjusting ring needs to be tightened; to do this you normally just need to remove the handle. 

Compression faucets are the most likely to drip.  The washers or seals of compression faucets are repeatedly torqued down against the metal valve seats, so they eventually wear out. When they do, the result is relentless dripping that can corrode fixtures, stain sink bowls, and waste a lot of water if they are not fixed.

"Washerless" is a termed used with disc, cartridge, and ball faucets because they don't utilize the same types of compression washers. They are far less prone to drip from the spout. They too have moving parts sealed by O-rings that wear out, though, evidenced by leaking from the base of the faucet itself.

Disc Faucet Repairs
Disc faucets leak when the inlet and outlet seals wear or when sediment builds up in the faucet inlets. Disc assemblies don't often wear out, but sometimes the inlet and outlet seals fail. Turn off the shutoff valve for the fixture and replace the worn seals with duplicates of the original ones (take the faulty ones with you when you buy replacements). Be sure to realign the seals on the bottom of the cartridge with the holes in the faucet.

Cartridge Faucet Repairs
Cartridge faucets leak when the O-ring cushioning the cartridge stem wears or breaks. When they drip it might mean the cartridge needs replacing. You can buy replacements at plumbing supply retailers, major home improvement centers, and some hardware stores. In many cases, they may need to be special ordered. Be sure the replacement matches the original.

Ball Faucet Repairs
Ball faucets such as those made by Delta and Price Pfister drip when the inlet seals wear and leak when the O-rings wear or break. If the handle leaks, tighten the adjusting ring after lifting the handle off the assembly. If the handle continues to leak, replace the cap; if the spout drips, the inlet seals and springs or the ball need replacing.

General Faucet Repair Tips
At the first sign of a drip, don't torque down on a faucet in an effort to stop the drip--in most cases, this will just damage the faucet. Instead, replace the washers or seals that are failing to seal off the faucet's flow. If you have an older faucet that is chronically leaky, consider replacing it with a new, high-quality model. See the HomeTips Bathroom Faucet Buying Guide or the Kitchen Faucet Buying Guide for unbiased advice.