There are several definitions of acids and bases.  The two most useful and most commonly encountered are the Lewis and Brønsted definitions.

According to Brønsted, an acid is a species that has a tendency to lose a proton, and a base is a species with a tendency to gain a proton.  According to Lewis, an acid is a species capable of accepting an electron pair, and a base is a species capable of donating an electron pair.

Therefore, it can be seen, that Lewis and Brønsted bases are in fact identical, as any species that gains a proton, does so by use of a pair of electrons.  However, on inspection of the definitions of acids, it is apparent that all Brønsted acids must also be Lewis acids, but that the reverse is not necessarily true. That is to say, Brønsted acids are a subset of Lewis acids.  An example of a Lewis acid that is not also a Brønsted acid is boron trifluoride:

Other common examples are aluminium chloride and zinc chloride.

Conjugate Acids and Bases

All acids and bases have their corresponding conjugate entities. ie:

In this scheme, HA is an acid, and A^-  is the corresponding conjugate base.  Similarly, B is the base, and its conjugate acid is BH⁺.  This is by virtue of the fact that the reaction is reversible, hence BH⁺ can act as an acid by donating protons to A^-