Let $A$ is a ring and $P$ is a prime ideal in $A$. What is the meaning of writing $PA_P$?
This object appears in the proof of Item (7.D) of Chapter 3 in Matsumura's Commutative Algebra.
I understand that $A_P$ denotes the localization of $A$ using the multiplicative set $A\setminus P$.
By writing $PA_P$ do we mean $$\{(pa)/s:\ p\in P, a\in A, s\in R\setminus P\}$$ which is same as the ideal generated by the extension of $P$ in $A_P$?
Is this correct?
Your interpretation is correct: $PA_P$ denotes the unique maximal ideal of $A_P$, i.e. the extension of $P$ to $A_P$ via the canonical morphism $A \to A_P$.
Do note that you can slightly simplify the definition as a set, though, because $pa \in P$ for every $p \in P$ and $a \in A$: $$ PA_P = \left\{\frac{p}{s} \in A_P : p \in P \,\text{ and }\, s \in A \setminus P\right\} $$