E2 is shorthand in organic chemistry for a particular mechanism of chemical reaction, so-called because it is an elimination (E) and is bimolecular (2).
An E2 reaction typically involves the addition of a base to a hydrocarbon in solution. The base removes a beta-proton, i.e. a hydrogen that is bonded to a carbon adjacent to the alpha-carbon. At the same time, this causes the breaking of the bond between the alpha-carbon and a leaving group and simultaneously the formation of a double bond, or pi bond, between the two carbons. Since all three things happen at the same time, E2 is called a concerted mechanism.
For example:
X H R H
| | \ /
R-C-C-R + OH- --> C=C + H-OH + X-
| | / \
H H H R
Where X is the leaving group, and R represents arbitrary carbon chains.
In general, the leaving group and the removed hydrogen must in an anti configuration across the carbon-carbon bond. (Meaning that if you looked down the carbon-carbon bond, they would be 180 degrees apart.) For chiral molecules, this implies a specific stereochemistry in the alkene product. For cyclic molecules, where groups cannot freely rotate around bonds, this may prevent E2 altogether.
In addition, E2 is a beta-elimination, so it can only occur when there are beta-protons. If a base is added to a compound with no beta-protons, alpha-elimination may occur instead. When there are multiple beta-protons, the more stable isomer of the product is formed in greater amounts.
For example, in the following reaction, the first product dominates because alkenes are stabilized by having carbons adjacent to the double bond.
H3C X H H3C CH3 (CH3)2HC H
| | | \ / \ /
H-C-C-C-H + OH- --> C=C + C=C + H-OH + X-
| | | / \ / \
H3C H H H3C H H H
E2 eliminations are usually in competition with SN2 nucleophilic substitutions. In general, SN2 is more easily inhibited by sterics, so E2 is more likely the more "crowded" the alpha carbon is, such as from branching at the alpha- or beta-carbons. | 
