| Zorn's lemma is a form of the axiom of choice which is technically very useful for proving existence theorems. For instance, it follows directly from Zorn's lemma that every ring has a maximal ideal and every vector space has a basis (algebraic, that is, Hamel basis). In some subfields of mathematics, arguments of this pattern are so common that they are referred to as zornification or zornication.
To rephrase BelDion's statement above a little: In a poset, if every chain has an upper bound, then the entire poset has a maximal element.
Some Polish mathematicians refer to this lemma as the Kuratowski-Zorn lemma, to properly credit its first appearance, in a paper of Kazimierz Kuratowski. For more information see Set theory for the working mathematician by Krzyzstof Ciesielski (London Mathematical Society student texts, Cambridge University Press). | 
