A nucleophile and a base are both nucleophilic that means they both have afinity for a electron poor target. Both nucleophile and base are electron rich, often have lone pair of electrons such as Oxygen and Nitrogen.
A nucleophile usually attacks electron poor atom, for example carbonyle atom, forms a new bond and forces another to bond to break, simply pushes out a leaving group. Typical nucleophile could be alcohol. On the other hand, base attacks an electron poor atom such as Hydrogen and binds with it, therefore takes it from the molecule that forms a carbanion. A typical base is for example triethylamine.