First of all, remember that basicity is asubsetof nucleophilicity. All nucleophiles are Lewis bases; they donate a lone pair of electrons. A “base” (or, “Brønsted base”) is just the name we give to a nucleophile when it’s forming a bond to a proton (H+). To summarize, when we’re talking about basicity and nucleophilicity, we’re talking about these two types of events.
Basicity: nucleophile attacks hydrogen
Nucleophilicity: nucleophile attacks any atom other than hydrogen. Because we’re talking about organic chemistry here, for our purposes, this is going to mean “carbon” most of the time.