An order (total order or partial order) is said to be well ordered (or a well ordering, sometimes even a well order, irregardless of the irreparable damage done to the English language) iff every set of elements has a minimal element.

For instance, the natural numbers are well ordered, but the rational numbers, real numbers, and integers are not. For partial orders, a hierarchy given by a (rooted) tree is a well order (since an element closest to the root in some set is minimal), even if the tree is infinite.

Having a well ordered set, certain induction-like techniques can be performed on it. Unfortunately, the only way to get a well ordering of a general set is to invoke the Axiom of Choice. So you get a very "unnatural" order, which will have virtually no properties in common with the problem domain. Nonetheless, it's better than nothing, and sometimes used.