Step 1: Understanding the formula.
The molecular formula C$_2$H$_7$N suggests that the compound contains 2 carbon atoms, 7 hydrogen atoms, and 1 nitrogen atom. This molecular formula corresponds to both a primary amine (1$^\circ$) and a secondary amine (2$^\circ$).
Step 2: Analyzing the options.
(A) only 2$^\circ$ amine: This is incorrect, as the formula allows for the possibility of a primary amine as well.
(B) 1$^\circ$ amine and 2$^\circ$ amine: This is correct, as C$_2$H$_7$N could correspond to both 1$^\circ$ amine and 2$^\circ$ amine structures.
(C) only 1$^\circ$ amine: This is incorrect, as the formula can also represent a secondary amine.
(D) 2$^\circ$ amine and 3$^\circ$ amine: This is incorrect, as the formula does not support a tertiary amine.
Step 3: Conclusion.
The correct answer is (B) 1$^\circ$ amine and 2$^\circ$ amine, as both types of amines can be represented by the given formula.