The number of possible organic reactions is basically infinite. However, certain general patterns are observed that can be used to describe many common or useful reactions. Each reaction has a stepwise mechanism that says how it happens, although this detailed description of steps is not always clear from a list of reactants alone.
Organic reactions can be organized into several basic types:
Here are some common reactions that you may run into in a college organic chemistry course: addition of HX to an alkene, halogen addition reaction, halohydrin formation reaction, oxymercuration reaction, hydroboration-oxidation reaction, hydrooxylation reaction, ozonolysis reaction, nucleophilic substitution, halogenation of an alkyl, hydration and dehydration.
Here is a series of reactions that a chemist can perform to change molecules in a defined way: Beckmann rearrangement, Friedel-Craft-Alkylation, Diels-Alder reaction, Pinner reaction, Sharpless epoxidation, Sharpless bishydroxylation, Swern oxidation
As seen here, specific reactions are sometimes named after the chemist who developed them.