I am busy studying Rosen's "Number Theory in Function Fields" Chapter 5. In this chapter Rosen defines a prime $M$ in the function field $K$ over $F$ to be a discrete valuation ring $R$ with maximal ideal $M$ such that $F\subset R$ and $R/M$ is isomorphic to $K$. He then states that $\mathsf{ord}_M(*)$ denote the ord function associated with $R$.
I know that for integers a given prime number $p$ we define $\mathsf{ord}_p(a)$ to be the largest number $n$ such that $a\equiv 0 \mod(p^n)$. Is there a similar definition for this $\mathsf{ord}_M(*)$ function that Rosen mentions?
First, when Rosen says the "quotient field" of $R$ is $K$, he means its field of fractions, not the residue field $R/M$. So every element of $K$ can be written as $a/b$ with $a,b \in R$.
The definition of $\newcommand{\ord}{\operatorname{ord}} \ord_M$ is really quite close to the definition for prime numbers that you've given. Since $R$ is a DVR, then $M$ is principal, so $M = (t)$ for some $t \in R$. Given $r \in R \setminus \{0\}$, then we can write $r = u t^e$ for some unit $u \in R^\times$ and some $e \in \mathbb{Z}_{\geq 0}$. We define $\ord_M(r) = e$ and extend the definition to $K \setminus \{0\}$ by letting $\ord_M(a/b) = \ord_M(a) - \ord_M(b)$.
This agrees with the definition for prime numbers you gave, since if $r = u t^e$ as above, then $r \equiv 0 \pmod{t^e}$, and this is the largest such nonnegative integer $e$. As an aside, if you don't want to choose a uniformizer $t$, $\ord_M(r)$ can be equivalently defined as the largest $e \in \mathbb{Z}_{\geq 0}$ such that $r \in M^e$, where we take $M^0 = R$.