Read the post on substitution reactions first, as many of the concepts follow on…
In an elimination reaction, fragments of a molecule are removed from adjacent atoms and replaced with a double bond. There are two mechanisms for an elimination, E1 and E2.
The E1 reaction starts in the same way as an Sn1 reaction: the removal of the leaving group to form a carbocation. The nucleophile can now either attach to the carbocation and reaction becomes a substitution, or it may act as a base and take one of the hydrogens attached to an adjacent carbon. If this occurs, the elimination product is formed.
E1 reactions will occur under similar conditions to Sn1: they must form a stable carbocation and have a good leaving group.
E2 reactions are bimolecular eliminations, with the elimination occurring in a single step
An E2 reaction requires the nucleophile to act as a base and taken a proton from a carbon. Alkyl hydrogens are not very acidic, so this reaction requires a strong base to occur.
In order to distinguish E1 from E2, you need to ask the same questions for Sn1 and Sn2, the only difference is that a strong base is required for E2.
There are situations where it is possible to get two elimination products, just as in addition reactions it is possible to place the hydrogen on two difference carbons. For elimination reactions, the most substituted product will be the major product, as the more substituted an alkene is, the most stable it is.