Why this happens

In a single-handle faucet, one cartridge under the handle controls both flow and temperature. Moen's own troubleshooting guidance for a spout drip points at the cartridge, and Pfister says most spout drips trace back to the cartridge under the handle. Seals inside the cartridge wear, and water seeps past them even with the handle off. Kohler notes a continuous leak usually means debris or wear in the valve or cartridge, while a leak that stops on its own is often debris sitting on a seal.

How to fix it

Shut off both stops under the sink and open the faucet to release pressure. Close the drain so small parts cannot fall in.
Remove the handle. Look for a set screw under a cap or on the side of the handle. Most take a small Allen key.
Unscrew the retaining nut or pull the retaining clip, then pull the cartridge straight up. Note which way it faced so the new one goes in the same way.
Before you buy anything, try Pfister's shortcut: wipe grit off the cartridge's gaskets with a damp cloth, reinstall it, reopen the stops slowly and test. If the drip stops, you are done. If not, shut the stops again, pull the cartridge, and match it to its replacement by brand and model number. If you cannot find a number, compare the length and the shape of the stem against the identification index linked below.
Seat the new cartridge in the same orientation, reinstall the clip or nut and the handle, reopen the stops slowly, and run the faucet for a minute to flush.

When to stop. If the stops under the sink will not close, or the cartridge will not come out with hand pressure and a puller, stop before you crack the faucet body.

The part that fixes it

Pick the cartridge that matches your valve, not your faucet's finish. The Moen 1225 fits most single-handle Moen sink faucets; the 35 mm and 40 mm ceramic cartridges fit the brands listed on each.