Partial Order
Definition 1. A relation $\leq$ on a set $A$ is called a partial order relation if and only if the relation $\leq$ is reflexive and transitive on $A$ and antisymmetric on $A$, that is, if $a\leq b$ and $b \leq a$, then $a = b$. A partially ordered set is a pair $(A, \leq)$, where $A$ is a set and $\leq$ is a partial order relation on $A$.
Total order
Definition 2. A total order relation $\leq$ on a set $A$ is a partial order relation such that for any pair of elements $a$ and $b$ in $A$, either $a\leq b$ or $b \leq a$. A totally ordered set is a pair $(A, \leq)$ where $A$ is a set and $\leq$ is a total order relation.
Well-Ordered
Definition 3. A totally ordered set $(A, \leq)$ is said to be well-ordered if and only if every nonempty subset $B$ of $A$ contains a unique minimal element; that is, if there exists an element $b \in B$ such that $b \leqq x$ for every $x \in B$. If $(A, \leq)$ is a well-ordered set, then the relation $\leq$ is called a well-order relation.