What does the quotient group $(A+B)/B$ actually mean? I understand that $A+B$ is the set containing all elements of the form $a+b$, wit $a\in A, b\in B$. 
When you do the quotient group, that's like forming equivalence classes modulo $B$. All elements in $B$ should be congruent to 0 modulo $B$, as far as I understand. So wouldn't $(A+B)/B$ be the same as $A/B$?
Where is the error in my reasoning?

No, but it is "the same" (isomorphic to) as $A/(A \cap B)$, that is, modding by the part of $B$ that lies inside $A$. Explicitly, we send:
$(a+b) + B\ (= a + B) \mapsto a + (A\cap B)$.
