Step 1: Understanding nitrogen sesquioxide.
Nitrogen sesquioxide is commonly known as N$_2$O$_3$. It is a nitrogen oxide compound where nitrogen and oxygen are present in a 2:3 ratio.
Step 2: Analyzing the options.
(A) N$_2$O: This is nitrogen monoxide, not nitrogen sesquioxide.
(B) N$_2$O$_3$: This is the correct answer, as N$_2$O$_3$ is the molecular formula for nitrogen sesquioxide.
(C) N$_2$O$_5$: This is nitrogen pentoxide, not nitrogen sesquioxide.
(D) N$_2$O$_4$: This is nitrogen tetroxide, not nitrogen sesquioxide.
Step 3: Conclusion.
The correct molecular formula is (B) N$_2$O$_3$.