Reusing aquarium gravel can promote a healthier aquarium. Gravel from an established aquarium can kick-start biological filtration. However, you do have to make sure you have the right gravel and it doesn't contain anything dangerous. With these caveats in mind, you can certainly reuse aquarium gravel or other substrate.
The Right Gravel
You need to make sure you have the right gravel for your tank before you reuse it. Most gravel sold at pet shops is epoxy-coated rocks. This gravel is chemically inert and does not cause problems. However, a few types of specialized aquariums use gravels that can cause problems in other tanks. For example, African cichlids prefer hard, alkaline water. Some gravel, like crushed coral gravel, slowly leaches calcium and other minerals into the water to achieve this. While this is great for an African cichlid tank, the same water could kill South American cichlids, which need soft, acidic water to thrive.
Adding gravel from another fish tank can greatly improve the health of an aquarium. In aquariums, a process called the nitrogen cycle breaks down fish waste into less toxic chemicals. Bacteria in the gravel carry out this process. Old, established aquariums typically have beneficial bacteria to spare. A few cups of gravel from an established aquarium can greatly accelerate the process of establishing these colonies, and get the tank ready for fish faster.
However, gravel can also introduce less friendly critters into an aquarium. In marine aquariums, pests like rock anemones may spread from one tank to another via gravel or decorations. Additionally, some pathogens like the protozoan that causes ich can live in aquarium gravel and spread to new tanks. Only use gravel from healthy aquariums, free of pests and disease. Otherwise, gravel could introduce dangerous organisms.
Some pet shops have capitalized on the beneficial properties of gravel and other substrates. Some pet shops sell -- or give away -- scoops of aquarium gravel or sand to help you get your beneficial bacteria established. For saltwater aquariums, this practice has been taken a step further. Some pet shops specializing in marine aquariums sell "live sand" -- sand that is already rich in nitrogen-eating bacteria to save you time in establishing your biological filter.