Alum is a group of chemicals, which in shaving circles are used for sealing nicks and cuts. The two most common forms are the styptic pencil and the alum block.
An alum block is used both to seal nicks and cuts, and as an after-shave treatment. The alum block is used by wetting it with cold water and rubbing it onto wet skin. The blocks are most commonly made from aluminum potassium sulfate, and are therefore milder than styptic pencils. They also take longer to seal cuts, therefore making them more useful as an aftershave treatment.
For sealing a cut, simply dab the block on the cut and rub it around a bit. Do note that it can take up to two minutes of direct application for the cut to completely seal.
As an after-shave treatment, rub the block over the wet skin twice and allow the skin to dry by itself. Afterward most razor-burn and small nicks should be gone. Do note that afterward your skin is going to be a bit more taut than normal, and your skin is going to feel a bit dry due to alum deposits. A remedy for this is to do a warm water rinse after shaving, then do the alum block treatment, and finish of with the normal cold water rinse. This eliminates the deposits to a certain degree, but the treatment will be milder. It's recommended that you experiment with it until you find a routine that works for you.
Remember that using an alum block can sting quite a bit, but it's a quick-fading pain that doesn't take long to disappear.
Rubbing your fingers on an alum block can improve your grip, making it easier to stretch the skin while shaving.