A well-order on a set $S$ is a relation $\prec$ that allows one to interpret $S$ as an ordinal number $\alpha$ and $\prec$ as the relation $\lt$ on the ordinal numbers less than $\alpha$. In particular, one can do induction on $S$ over $\prec$ (although the more general well-founded relations also allow this).
The well-ordering theorem states precisely that every set may be equipped with a well-order. This theorem follows from the axiom of choice, and is equivalent to it in the presence of excluded middle.
A binary relation $\prec$ on a set $S$ is a well-order if it is transitive, extensional, and well-founded. A set equipped with a well-order is called a well-ordered set, or (following ‘poset’) a woset. Actually, the term ‘well-ordered’ came first; ‘well-order’ is a back formation, which explains the strange grammar.
Other definitions of a well-order may be found in the literature; they are equivalent given excluded middle, but the definition above seems to be the most powerful in constructive mathematics. Specifically:
The really interesting thing here is that every well-order is linear; it is a constructive theorem that every linear order is weakly extensional (and so extensional if well-founded) and transitive. (For a weak counterexample, take the set of truth values with $x \prec y$ iff $y$ is true and $x$ is false; this is a well-order that's linear iff excluded middle holds.) For the other equivalences, we're simply using well-known classical equivalents for well-foundedness and the classical correspondence between a linear relation $\prec$ and its reflexive closure? $\preceq$.
For reference, a classical well-order is any order satisfying the last definition (a total order that is classically well-founded). A classically well-ordered set is a choice set, and so if any set with at least $2$ elements has a classical well-order, excluded middle follows.
As stated above, well-founded extensional transitive relations $\prec$ on a set $X$ are linear, assuming classical logic.
Order $X \times X$ lexicographically: $(a, b) \prec (a', b')$ if either $a \prec a'$ in $X$ or $a = a'$ and $b \prec b$ in $X$. It is not hard to see that the lexicographic order is well-founded (and in fact a well-order, although we do not need this). Now let $A \subset X \times X$ be the set of pairs $(x, y)$ of non-equal elements $x$ and $y$ that are incomparable in $X$, and suppose $A$ is inhabited. Then $A$ has a minimal element $(a, b)$ (using excluded middle). Then, for every $x \prec b$, either $a \preceq x$ or $x \preceq a$. If the former holds for some $x$, then $a \prec b$ follows by transitivity, contradiction. Hence $x \prec a$ for every $x \prec b$. Now let $a'$ be minimal such that $x \prec a' \preceq a$ for every $x \prec b$.
Claim:
We know already the right side is contained in the left. In the other direction, suppose $x \prec a'$. Since $x \prec a$ and $(a, b)$ was chosen minimal in the lexicographic order, $x$ and $b$ are comparable. If $b \preceq x \prec a'$, this contradicts minimality of $a'$. Thus $x \prec b$, i.e., the left side is contained in the right.
But now, by extensionality, $a' = b$, whence $b \preceq a$, contradiction. Therefore $A$ was empty, so that $X$ is connected and therefore (being already transitive and irreflexive and using excluded middle again) a linear order.
Any finite linearly ordered set $\{x_1 \lt \cdots \lt x_n\}$ is well-ordered.
The set of natural numbers is well-ordered under the usual order $\lt$.
More generally, any set of ordinal numbers (or even the proper class of all ordinal numbers) is well-ordered under the usual order $\lt$ (which, constructively, may not be linear).
The cardinal numbers of well-orderable sets (the well-orderable cardinals), forming a retract of the ordinals, are well-ordered. So by the well-ordering theorem, the class of all cardinal numbers is well-ordered.
A special case of the well-ordering theorem is the existence of a well-order on the set of real numbers; this is enough for many applications of the axiom of choice to analysis.
Any well-ordered set $S$ defines an ordinal number $\alpha$ and an order isomorphism $r$ between $S$ and the set of ordinal numbers less than $\alpha$; as such, $S$ may be identified (up to isomorphism of wosets) with the von Neumann ordinal $\alpha$. The idea is that the minimal element $\bot$ of $S$ itself (if any) is mapped to the ordinal number $0$, the minimal element of $S \setminus \{\bot\}$ (if any) is mapped to $1$, and so on; after which the next element of $S$ (if any) is mapped to $\omega$, and so on; and so on.
This may be defined immediately (and constructively) as a recursively defined function from $S$ to the class of all ordinal numbers:
the validity of this sort of recursive definition is precisely what the well-foundedness of $\prec$ allows. Here, $\beta^+$ is the successor of the ordinal number $\beta$, and $\sup$ is the supremum operation on ordinal numbers (which is the union of von Neumann ordinals).
Since $S$ is a set, the image of $r$ in the class of all ordinals is also a set (by the axiom of replacement), and one can now prove that $r$ is an order isomorphism between $S$ and the set of ordinals less than the next ordinal, $\alpha \coloneqq (\im r)^+$.
Given two well-ordered sets $S$ and $T$, a function $f\colon S \to T$ is a simulation of $S$ in $T$ if
Note that any simulation of $S$ in $T$ must be unique. Thus, well-ordered sets and simulations form a category that is in fact a (large) poset, in fact the poset of ordinal numbers.
A well-ordered set $S$ comes equipped with a successor, which is a partial map $succ\colon S \to S \,$ that sends $a \in S$ to the lowest element of the subset $S_a := \{ s \in S, a \prec s\}$, whenever this set is inhabited.
A limit well-order is a well-order $S$ whose successor map is a total function.
Similarly, one may define a successor functor on the category of well-ordered sets, taking $S$ to the well-order obtained by freely adjoining a (new) top element to $S$. Since this category (which is thin) can be regarded as itself a well-ordered proper class, this is a special case of the successor operation above. (Hence the ordinal of all ordinals is a limit ordinal.)