Rinsing or dousing the area immediately in cool or cold water is the best method for removing blood stains from a mattress or other fabric item. Blood easily dissolves in water, but it needs to be cold; hot water will "cook" the stain right into the fabric.
Cool water can be dabbed onto the mattress with a clean cloth rather than poured. Once the stain has been rinsed out, the mattress should be air-dried. Any application of heat will set any remaining blood into the mattress.
If all the blood has not been removed with water, both household ammonia and hydrogen peroxide (a common first aid item) can also be used to augment cleaning power. Hydrogen peroxide can be used at full strength and simply poured onto the mattress. Ammonia, in contrast, can only be used safely in solution; 1 tablespoon of ammonia to 1 cup of water is a good rule of thumb. Either one can be applied to the mattress and left to soak for about a half an hour. After that period, they should be rinsed out (or dabbed out with a cloth) with cool water.
Older stains can be removed using a paste made of half a cup of cornstarch, one tablespoon of salt, and a quarter cup of hydrogen peroxide. It should be left on the stain for about half an hour until fully dried and then scraped off.