We finished math camp by proving Sperner's Lemma to in turn prove Brouwer's Fixed Point Theorem.
Now, I certainly can't reproduce either of these proofs from scratch, though I can follow them. However, it was first proved in 1924, making it the second-oldest result whose proof I understood. The oldest would be proving that the set of differentiable functions is meager in the space of continuous functions, which stems from 1931.
Curse that the course in math history was never offered during my undergraduate years. Oh, and before I get too big of myself, Sperner proved it when he was 19.