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 base, by one pretty good definition, is something that is good at donating its electrons, allowing it to accept H+ - a proton. A nucleophile is a bit trickier to explain - it is good at donating its electrons, allowing it to react with certain species called electrophiles, forming a bond. The groups often overlap - hydroxide is a good base and a good nucleophile, as is methylamine. Iodide is a terrible base, but it's a great nucleophile.
Oftentimes, you want a good base, but a bad nucleophile. For instance, if you have a reaction that releases protons, you might want to mop up those protons as the reaction goes along. Many bases will interfere (Because of their nucleophilicity), but ones that are "sterically hindered" (crowded, pretty much), tend not to. DBU is one such base. Great base, terrible nucleophile.
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In freshman orgo, the best definition I've heard for nucleophilicity versus basicity is that nucleophilicity is kinetic, while basicity is thermodynamic.
Consider Proton Sponge and Hydride Sponge. Their outrageous property boost arises from relaxation of orbital steric interaction by grabbing a proton or hydride, respectively. The kinetics are slow but the grab has fangs.
Carbon-centered stable anions are easy: malonate, Meldrum's acid, tri(carboxymethyl)methide; tricyanomethide, orthothioester anions, etc. What happen if Hydride Sponge enters the solution? Its borons chelate anionic carbon to gain a 1/2 negative charge each, and a stable pentacoordinate carbon slowly appears. Worth a look with an inert counter-cation (crypt ligation?).
Proton Sponge and carbocations (guanidinium) might be nice. The somewhat unlikely Big Kahuna is both Sponges plus TCNQ to obtain stable cationic and anionic five-coordinate carbons in one molecule.
All I know is that a base attacks a PROTON (H+) and a nucleophile attacks an electron-deficient CARBON. I think that's the basic difference between a base and a nucleophile in organic chemistry.
Saw some nice chemistry the other day where they used DBU on a dibromoalkane to get the alkyne. really nice