Water heaters come equipped with a temperature and pressure relief valve, also known as a TPRV. This valve allows water or steam to escape from the water heater if the temperature or pressure gets too high. This prevents a water heater from exploding or turning into a missile.
When a TPRV at a water heater leaks, it's usually a simple fix; just replace the valve. These valves cost less than $15 and replacing the valve is a basic job. There are plenty of videos on YouTube showing how to do this, and the whole project probably won't take more than fifteen minutes.

Unless it starts leaking again.
If you replace a TPRV and it starts leaking again, it probably means the valve is doing its job; it's relieving excess pressure. When this happens, the fix gets a little bit more involved, and some troubleshooting is in order.

Verify the proper valve is installed
The TPRV on a water heater is set to discharge at 150 PSI or 210° Fahrenheit. These numbers will be printed right on the valve, or on a tag attached to the valve. See below.

If a pressure relief valve for a boiler is accidentally installed on a water heater, it will leak like crazy from the start. These valves may look identical, but they're set to go off at 30 psi, not 150 psi.
If a new TPRV is properly installed and it still leaks, you probably have a pressure problem.
Excessive pressure

If the pressure in a plumbing system ever exceeds 150 PSI, the TPRV will leak. Getting to the bottom of this issue should be quite simple and straightforward. Buy a pressure gauge with an extra indicator to show surges, and connect it to the plumbing system. It doesn't matter if it's connected to a hot or cold water pipe, because both will be at the same pressure.