This particular type of water torture is likely due to a failed washer inside a handle. The faucet is just the messenger.
To replace the washer, turn off the water supply valve under the sink. Stuff a rag in the drain so you don't lose parts, then take the handle apart. Pop the screw cover on top, remove the screw, and pull off the handle. Use a wrench to disassemble the stem, and line the parts up on the counter in the order they came off, so you know how it goes back together. Examine rubber parts or plastic cartridges for cracks, and take the offending piece to the hardware store for an exact replacement. Reassemble the parts you've laid out, in reverse. Then revel in the ensuing peace and quiet.
For more detailed diagrams, check out Fixing a Leaky Faucet