Explain this math thing like IAMA

I've been trudging through Hatcher's text on algebraic topology, and have come to the section on the Universal Coefficient Theorem for Cohomology, where "Ext" is introduced. I'm more curious about Ext from a homological algebra standpoint (as well as Tor): what do they do algebraically?

I've taken a couple of courses in algebra, a course in advanced linear algebra, a Galois theory course, and have some self-taught topology and algebraic topology. Let's hear it.