First, I give definitions and mathematical preliminaries. Secondly, I give a history of the derivation of Einstein's field equations. From this basis, I present a derivation of Schwarzschild's solution. A discussion then follows of various types of black holes: stationary, charged, rotating, and charged/rotating. I also give a pictorial representation of the properties of each algebraically special solution. Finally, I present a general definition of singularities along with a discussion of closed trapped surfaces and naked singularities.