A toilet that never quite stops running is usually the fill valve, and the part costs about eight dollars. This is one of the few plumbing jobs with no soldering, no glue and no way to flood the house, which makes it a good first repair if you have never opened a toilet tank.
Budget half an hour. Most of that is draining and sponging the tank.
Fill Valve or Flapper?
These two get mixed up constantly and they fail differently.
The fill valve is the tall column on the left side of the tank that refills it after a flush. When it fails, the tank keeps trickling water in and never shuts off, and it often whistles while filling.
The flapper is the rubber seal at the bottom. When it fails, water leaks from the tank down into the bowl, so the tank slowly empties and the toilet refills by itself every few minutes.
If you are hearing a phantom refill with no obvious running, start with the flapper. If the fill never stops, you are in the right place.
Which Valve to Buy
The Fluidmaster 400A fits nearly every standard American toilet and is what most hardware stores stock. There is a quieter PerforMAX version if the noise is your main complaint, and combo kits that bundle a flapper if you would rather replace both while the tank is open.
Measure nothing before you buy. These are universal and adjust to height during installation.