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).