Dimethylaminopyridine, or DMAP, finds general use in organic synthesis as a "nucleophilic catalyst." Its strongly donating dimethylamine group affords it substantial nucleophilicity.
The adducts formed by its nucleophilic addition, however, are quite reactive - analogous to an acid chloride - so DMAP can push a reaction along nicely. See the Wikipedia entry for an example mechanism.
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Basicity and nucleophilicity are two related concepts, but they don't always correlate. This is part of what makes teaching and learning chemistry so tricky, especially at first, when it seems like you're just learning a collection of facts (rather than the holistic wonder that is chemistry!).
A lot of reactions with nucleophiles' rates are determined by how good a leaving group you have. For leaving group reasons and others, DMAP is a great organocatalyst:
DMAP is one of the better nucleophilic catalysts - it's cheap, readily available, and works. Perhaps the one disadvantage is that it can show up on TLC plates under the UV, so it's best to use a standard dilute solution of DMAP if you're tracking the progress of an unknown reaction.