There are several steps involved for cleaning a mattress. First, you need to clean the spot. Ideally, you should do this as soon as possible, do not wait to clean it. The longer you wait, the more it will soak through the mattress and into inside layers you cannot easily clean.
First, blot to absorb any dampness using a terry cloth or paper towels. Then using water, clean an area 4″ larger than the spot, then absorb the water. Repeat several times.
If there is an odor associated with it (such as cat urine) you will likely need to use an enzyme cleaner. Follow instructions on the bottle.
Always be sure the mattress is completely dry before flipping it over or replacing the sheets. Opening windows, using a fan or heater, or allowing sunshine into the room can help dry the mattress faster.