If you are considering the Bronsted-Lowry theory, then an acid is defined as a "proton donor". A proton is a H+, which would draw electrons to it. A base is defined as a proton acceptor. Using similar reasoning, a base would either be a negatively charged ion (anion) or a species with one or more unshared electron pairs. Having 'extra' electrons would (sort of) make it a 'donor' of electrons.
However, we rarely look at electron donation and/or accepting in the B/L theory. But the Lewis theory does - just opposite of the way you're stating it in the question. In the Lewis theory an acid is defied as an electron pair acceptor (thus, H+ or any species that has less than an octet of electrons, such as BCl3), and a base is defined as an electron pair donor (thus, any anion or any species with unshared electron pairs, such as NH3)