What is the difference between total order relations and well order relations?

Question

I know it has to be a partial order relation in order for it to be a well order relation or total order relation, but what are the differences between them.

Answer 1

From Wikipedia:

In mathematics, a well-order (or well-ordering or well-order relation) on a set $S$ is a total order on $S$ with the property that every non-empty subset of $S$ has a least element in this ordering.

In particular, a well-order is a total order (which itself is a partial order).

But the other implication is not true: for instance, take the (natural) order on $\mathbb{Z}$. It is a total order, yet not a well-order — since in particular $\mathbb{Z}$ has no least element.