Well-order

Definition

Let $A$ be a set. A well-order on $A$ is a total order $\le$ on $A$ such that each nonempty subset of $A$ has a least element. The pair $(A, \le)$ is a well-ordered set.

Remarks

The usual order on $\mathbb{N}$ is a well-order. The usual orders on $\mathbb{Z}$ and on $[0, 1]$ are not well-orders: $\mathbb{Z}$ and $(0, 1]$ have no least element. A total order is a well-order exactly when its strict order $<$ is well-founded. The well-ordering theorem says that each set has a well-order.

Explanations

L1 human

In a well-ordered set, each nonempty part has a first element. So you can always say "take the smallest element with this property".