Enzymes are what are called 'biological catalysts;' that means they speed up how fast a reaction takes but they are unchanged by the reaction themselves. They work by lowering the activation energy needed for two molecules to react, that is how much energy is needed to make the molecules react. They do this by bringing the two molecules together (what we call substrates) and making the transition states of the reaction more energetically favourable. In other words, when we react say A with B to get E, we have to go through a series of steps first, so A and B makes C, which then reacts to form D which reacts to form E. Enzymes make this reaction require less energy.
Enzymes are proteins and have a very specific structure with an area on them called the active site. This is specific to a particular substrate (think like a lock and key where the key is the substrate and the lock is the enzyme). Once the substrate binds to the enzyme's active site, the active site changes its shape slightly to fit around the substrate. After the reaction has taken place, the product(s) of the reaction are released from the active site.