Enzymes are proteins that act as catalysts of specific reactions. By providing an alternative reaction with a lower activation energy, they allow the reaction to proceed at a much higher rate. It is important to note that enzymes do not change the equilibria of a reaction; they can only increase the rate. Without the enzyme, the reaction would therefore still proceed in the same direction; however, it would be slower, often a lot slower. Enzymes usually increase reaction rates between 105 and 107 times. Enzymes are required in our body to perform specific metabolic reactions. They are highly specific for their substrates, much like a key is specific to a specific door. The specificity of enzymes for their substrates led Emil Fischer to propose the so-called "lock and key" hypothesis in 1894. The "lock and key" hypothesis implies that enzymes are static molecules, which they are not. Another mechanism called induced fit is the preferred model; according to this model, the enzyme undergoes conformational changes when binding to the substrate, and these changes are necessary for catalysis [1,2].
Figure 1: A reaction from a substrate to product is a transition from one energy state to another. A transition state exists between the substrate and product. This state has a higher energy level than both the substrate and product. A catalyst will lower this energy level so that the transition energy is reached more easily, and this results in a faster reaction.
There are many different ways by which enzymes enhance the rate of their specific reactions. When two substrates are transformed into one or more products, the enzyme will bind both substrates, thereby ensuring that they are in close vicinity and correctly oriented towards each other. When the reaction occurs without the enzyme, the 2 substrates have to hit each other from the correct angle and with enough energy (speed) to overcome the much higher activation energy. When the substrates are bound by an enzyme, these challenges are overcome by the structure of the enzyme. An important characteristic of enzymes (and catalysts in general) is that they are not used up during the reaction. When a reaction has been catalyzed by the enzyme, it is ready to convert another set of substrates. 
Lehninger, Albert L.; Nelson, David L.; Cox, Michael M. (2008). Principles of Biochemistry (5th ed.). New York, NY: W.H. Freeman and Company. ISBN 978-0-7167-7108-1.
Atkins, Peter W.; de Paula, Julio; Friedman, Ronald (2009). Quanta, Matter, and Change: A molecular approach to physical chemistry. Oxford University Press. ISBN 978-0-19-920606-3.