I'll do an AoC this year: each day post another statement equivalent to the Axiom of Choice.

In these posts "P is equivalent to axiom of choice" will have its standard meaning of "ZF ⊢ P → AC; ZFC ⊢ P", using classical logic.

My first post each days will contain the statement equivalent to AC. You are encouraged to try to write the equivalence proofs yourself, I'll give my own proof in the replies if requested, so beware spoilers.

Zorn's Lemma is a formulation of the axiom of choice that looks very abstract, but in my experience is the most useful in practice.

The statement goes:
If X is a partial order such that each chain of X has an upper bound in X, then X contains a maximal element.

(A chain of X is a subset C ⊆ X, such that all c₁, c₂ ∈ C are comparable: c₁ ≤ c₂ or c₂ ≤ c₁.)