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Zorn's lemma states: if (P, ≤) is a non-empty partially ordered set in which every chain (totally ordered subset) has an upper bound in P, then P has at least one maximal element. It is equivalent to the axiom of choice over ZF and to the well-ordering theorem. Zorn's lemma is the preferred formulation of choice in algebra and analysis: it directly produces maximal objects such as maximal ideals in rings, Hamel bases for vector spaces over arbitrary fields, and ultrafilters. Its power lies in converting the global choice principle into a local maximality argument that is easy to apply in specific algebraic or topological settings.
Memorize the precise statement: every chain has an upper bound (not necessarily a maximum), and the conclusion is a maximal element (not a maximum of P). Apply it to produce: (1) a maximal ideal in any non-trivial commutative ring, (2) a basis for any vector space, (3) a maximal consistent set of formulas. In each case, identify the poset P and verify chains have upper bounds.
You already know what a partial order is: a relation ≤ on a set P that is reflexive, antisymmetric, and transitive, but where some elements may be incomparable. You also know the axiom of choice in its set-theoretic form: for any collection of non-empty sets, there exists a function that selects one element from each. Zorn's lemma is neither of these things — it is a theorem equivalent to the axiom of choice, but formulated in the language of partial orders where it is far easier to apply.
The key vocabulary: a chain in P is a totally ordered subset — every two elements in a chain are comparable. An upper bound for a chain C is an element u ∈ P (not necessarily in C) such that c ≤ u for every c ∈ C. A maximal element m is an element such that nothing in P strictly exceeds it: if m ≤ x then m = x. Note carefully the asymmetry: "upper bound" is a condition on chains, "maximal" is a condition on the whole set P. Zorn's lemma says: if every chain has an upper bound, then P has a maximal element. This is the conclusion — the existence of at least one m with nothing above it.
Why is this useful? The power of Zorn's lemma is that it converts the abstract axiom of choice into a concrete existence proof. Here is the standard template: (1) form a poset P whose elements are the objects you want to exist (e.g., consistent sets of formulas, ideals in a ring, linearly independent subsets of a vector space), ordered by inclusion; (2) verify that every chain in P has an upper bound (usually the union of the chain); (3) conclude by Zorn's lemma that P has a maximal element; (4) show that maximality forces the element to have the desired property (e.g., a maximal linearly independent set is a basis). The lemma handles the existence; you handle the characterization.
The maximal vs. maximum distinction is critical. A maximum of P is an element that is above *every* element of P — there can be at most one. A maximal element is one that has nothing strictly above it, but there may be many maximal elements that are incomparable to each other. Consider the poset of proper subsets of {1, 2, 3} ordered by inclusion: {1, 2}, {1, 3}, and {2, 3} are all maximal (no two-element subset contains another two-element subset), but none is a maximum (none contains the others). Zorn's lemma guarantees at least one maximal element when the chain condition holds — it says nothing about uniqueness or about a maximum.
The equivalence to the axiom of choice and the well-ordering theorem is a deep result: over ZF (Zermelo-Fraenkel set theory without choice), these three statements are mutually derivable. In practice, Zorn's lemma is the workhorse version of choice in algebra because it speaks directly in terms of the algebraic structures at hand. When you encounter a proof that "every non-trivial commutative ring has a maximal ideal" or "every vector space has a basis," the proof almost certainly invokes Zorn's lemma — and now you have the framework to see exactly why that works.