There are two main ways to do an elimination reaction. The first is called E1. It happens in two steps. If there is a good leaving group in the molecule, it can leave on its own. Then, a base or another substance can take away the other group. The second way is called E2. This happens only in one step. In this case, the base takes away one group while at the same time the leaving group goes away.
A common elimination reaction happens with alcohols. If there is strong acid around, a molecule of water can be eliminated to form the double bond.