What does a "well ordering on $\Bbb R$" mean?

Question

A well order is a poset such that every non empty subset of this set has a least element. Does "well ordering on $\Bbb R$" mean that every non empty subset of $\Bbb R$ (which is a poset) has a least element?

Answer 1

A well-ordering on $\Bbb R$ is a binary relation $\preceq$ between elements of $\Bbb R$ such that $(\Bbb R,\preceq)$ is well-ordered. I.e. a relation $\preceq$ which is reflexive, antisymmetric, transitive and such that every non-empty subset $S\subseteq\Bbb R$ has an element $x\in S$ such that, for all $y\in S$, $x\preceq y$.

Answer 2

It is a theorem of set theory that every set can be endowed with a well-ordering (Zermelo's theorem).

Actually, the axiom of choice, Zorn's lemma and Zermelo's theorem are equivalent.