How is the quotient of two ideals defined?

Your teacher is thinking to the ideal $J/I$ of $A/I$.

There is a surjective ring homomorphism $A\to A/I$, sending $a$ to $a+I$. The image of $J$ under this homomorphism is an ideal of $A/I$ and it is quite naturally denoted $J/I$, because it consists of all elements of the form $x+I$, with $x\in J$. This assumes that $J\supseteq I$.

In the general case, if $J$ is any ideal of $A$, its image in $A/I$ under the canonical homomorphism above is $(J+I)/I$.

For instance, $\sqrt{I}/I$ is the nilradical of $A/I$. Indeed, if $x\in\sqrt{I}$, then $x^n\in I$ for some $n$, so the element $x+I$ is nilpotent in $A/I$. The converse is also clear.

If you like you can just say that $J/I$ is the quotient of abelian groups.

Beyond that, it still has a left and right $R$ bimodule structure inherited as a submodule of $A/I$.

It's also consistent to think of it as a quotient of a ring not necessarily having identity.

Even if $I$ isn't a ring with identity, nothing goes wrong. All of these nice structures are still there.