Without the nitrogen cycle, it would hard to keep fish alive in aquariums. In the nitrogen cycle, bacteria eat toxic ammonia from fish waste and break it down to less toxic nitrogen-containing chemicals. Establishing colonies of this bacteria is called "cycling" an aquarium. There are ways to speed the process.
Fishless cycling is the process of cycling an aquarium using rotting fish food or liquid ammonia instead of hardy aquarium fish. In normal fish tank cycling, you typically add your fish in several groups to give the beneficial bacteria time to multiply. However, in fishless cycling, you can add all of your fish at once, after several weeks of cycling an empty tank. The biggest drawback is staring at an empty tank while it cycles without fish. This process takes less time overall than traditional cycling.
Silent cycling is another alternative to traditional aquarium cycling. In silent cycling, you plant an aquarium as densely as you can. Plants metabolize nitrogen compounds and can reduce ammonia levels enough to allow you to cycle a tank more quickly. The biggest drawback to this method is that aquarium plants require their own specific care. If you don't know how to keep them happy, they will die, rot and add even more ammonia to the aquarium water.
Several products are on the market that claim to help speed the nitrogen cycle. They may or may not work. Most supposedly contain nitrifying bacteria to speed the process of cycling an aquarium. However, the specific species of bacteria that perform in the nitrogen cycle have recently come into question, making the claims on the packaging somewhat dubious. At the very least, these products probably won't hurt anything. And they could possibly help.
There are several ways to speed the nitrogen cycle. You can "seed" a tank with gravel or decorations from an established, cycled aquarium. In saltwater aquariums, live sand and live rock are sand and dead coral skeletons rich with ammonia-eating bacteria. Additionally, nitrifying bacteria thrive at a pH between 7.5 to 8.0 and at temperatures between 77 to 86 degrees. If your fish can handle these conditions, provide them. Good current also helps nitrifying bacteria to thrive.