Worn cartridge or washer
The valve no longer seals, so the spout drips after it is closed.
Drips from the spout after it is turned off, water around the handle or a puddle in the cabinet below.
Short answer
Where the water appears tells you the cause. Drips from the spout mean a worn cartridge or washer inside the valve. Water around the handle or base means O-rings. Water under the sink means a supply line, valve or drain connection rather than the faucet itself.

The valve no longer seals, so the spout drips after it is closed.
Seals around the stem or spout let water seep out when the faucet is on.
Water runs down the supply line and pools in the cabinet.
Water leaks from the drain body or trap when the sink drains.
A dripping faucet looks small, but it adds up over weeks and scale builds on the valve seat while it drips. Most are fixed with a new cartridge or washers. When the body is corroded or parts are discontinued, the plumber may suggest a new faucet instead. The repair page has more detail. Two-handle faucets have two sets of parts, so a drip may come from one side only. Closing one shut-off valve at a time shows which side is leaking, which tells the plumber which cartridge or washer to bring. Photos of the handles help too.
Call when a drip continues after the handle is fully closed, water collects in the cabinet, the shut-off valves will not close, or parts for your faucet are hard to find.
A cartridge or washer worn past the point of sealing, often after scale builds up.
O-rings around the stem or cartridge have dried out or worn.
Usually a supply line, valve or drain connection rather than the faucet itself.
A steady drip adds up over time. It is worth fixing for that reason alone.
Not usually, unless water is reaching the cabinet floor. Fix it soon to avoid damage.
No. That damages the cartridge or seat and makes the drip worse.
The faucet brand, how many handles it has and photos of the faucet and the space under the sink.
No. Cartridges are specific to the brand and model, which is why identifying the faucet helps.
The hot side cartridge or washer is worn. Repairing that side usually stops it.
Yes. Pressure above about 80 psi strains cartridges and washers.