There are four major categories into which organic reactions fall. These are:

1. Substitution
2. Addition
3. Elimination
4. Rearrangement

Substitution

To deal with 1. first, substitution usually occurs on carbon.  There are two sub-categories of this type: nucleophilic, and electrophilic.

This is an example of electrophilic substitution (nitration of benzene):

Below is a characteristic example of nucleophilic substitution:

NC^- is the nucleophile in this case.

Addition

Addition reactions are also sub-categorised into nucleophilic and electrophilic reactions.  They can also be radical.  For example, the addition to C-C double bonds is usually an electrophilic or radical process. e.g.:

This can either be initiated by attack of H⁺ or Br^- on the double bond.  However, the addition reactions to carbonyl groups (e.g. in either aldehydes or ketones) are nucleophilic. e.g.:

Elimination

Elimination reactions are essentially addition reactions running backwards. e.g.:

Rearrangements

Rearrangements can occur via intermediates that closely resemble cations, anions or radicals.  However, normally we find that rearrangements occur via carbocations.  During a rearrangement, as the name indicates, it is possible to change the carbon skeleton significantly. e.g.: the Pinacol rearrangement

All these reactions will be dealt with in more detail elsewhere.