Chemical reactions occur whenever bonds are formed or broken between molecules. Why certain atoms combine with which other atoms is a complex question which is explained exhaustively by quantum chemistry. The simple version is that atoms bond together electromagnetically based on the properties of their electron shells. There are various types of chemical bonds; in general, the more closely overlapping the electron shells are, the stronger the bond is. Chemical reactions happen when heat, radiation, and/or foreign chemicals disturb an equilibrium condition and cause the breakage and/or formation of chemical bonds, thereby giving rise to new molecular forms.
Chemical reactions can be classified into a few categories. The simplest is probably synthesis, where two or more molecules or atoms combine into a new molecule. For example, iron plus oxygen forms iron oxide, or rust. The opposite of synthesis is analysis, or chemical decomposition, where a molecule breaks apart into its constituents. This occurs when the electrolysis of water produces oxygen and hydrogen gas. Substitution, another variety of chemical reaction, happens when a more reactive atom or molecule ejects a part of another molecule and takes its place. This happens whenever an acid eats into a metal. And finally, there is combustion, which occurs when something like wood undergoes oxidation and is incinerated.
The number of possible chemical reactions in nature is extremely huge. There are more than billions - because very large molecules have so many components, the number of possible reactions is immense. This is particularly true in organic chemistry, where million-atom molecules are the norm. Life is possible because of the tremendous amount of possible chemical configurations achievable by organic molecules.
In organics especially, special molecules called enzymes are capable of accelerating chemical reactions without themselves undergoing any chemical change. Another word for this is a catalyst. Without catalytic enzymes, the everyday cascade of the numerous reactions that make up life would take too long to be practical. In cars, catalytic converters process exhaust from the engine to make it far less toxic than it would be otherwise. A very important task of chemists is to thoroughly understand catalysts.
It is impossible to understate the importance of chemical reactions in the world we live in. Numerous scientific advancements occur when we understand even just a little bit more about chemistry. Perhaps one of the most important to human civilization is the Haber-Bosch process - a chemical reaction where nitrogen from the atmosphere is compressed and combined with other chemicals in a superheated environment to create artificial fertilizer. Without it, we wouldn't be able to grow all the plants we use to feed ourselves, and the world population would be far smaller and more hungry.