Hygrometers measure the humidity that is in a humidor.
If the humidity within a humidor is too low or too high, than cigars can become damaged. Hygrometers will help you maintain an appropriate level.
If hygrometers show a high level of humidity in a humidor, than cigars may become moldy.
If a hygrometer shows a low level of humidity within a humidor, than cigars may dry out and crack.
Since hygrometers are an essential part of humidors, many humidors come with them built in. There are also many hygrometers that are available without a humidor in case yours breaks.