A base in the sense of Sn1, Sn2, E1, E2 are categorized by the ability to pull of hydrogens. Right now, you will probably see only a few common strong bases. They would be lithium diisopropylamide, n-butyllithium, um...yea. Strong bases are ones that have a negative charge that is destabilized, or very strong, with lots of electron donators (however, I think most of that will be over your head).
Strong nucleophiles are those that have a large-polarizable electron shells that can easily penetrate to attack.