Motivation of the definition of Euclidean Domain We know the definition of Euclidean Domain is

An Euclidean Domain is an Integral domain $(E,+,\ast )$ together with a function $v: E\setminus \{0\} \to \mathbb{N} \cup  \{0\} $ such that
(i) for all $a,b \in E$ with $b \neq 0$, there exist $q,r \in E$ such that $a = qb + r$ ,where $r=0$ or $v(r) \lt v(b)$
(ii) for all $a,b \in E \setminus \{0\}$, $v(a) \leq v(ab)$.

What is the motivation behind the definition of Euclidean Domain. Is it a generalisation of something?

Yes, in fact it's generalization of division. In integers we can divide any integer $a$ to to any other like $b$ and write $a=qb+r$ that $q$ is quotient and $r$ is reminder that less than $b$ always. Now we generalize it to Euclidean domain by that definition and function.
The second criteria is generalization of $a \leq ab$.
Also the "order" is actually generalized before this.
