What is the main difference between finite fields and rings? In the course I'm studying, if I've understood it right, the main difference between the two is supposed to be that finite fields have division (inverse multiplication) while rings don't. But as I remember, rings also had inverse multiplication, so I can't see any difference.

What is the main difference between finite fields and rings?
Since you specified finite fields and other answers didn't talk about it, I am adding the following:

Fields are rings which are commutative and in which all nonzero elements have multiplicative inverse.

But finite fields have another important property that distinguish them from rings: every finite field is completely specified by its order, because they always have exactly $p^n$ elements for some prime $p$ and natural $n$ and any two finite fields of same order are isomorphic. 

Notice, saying that two fields are isomorphic means that they are the same, but represented in two different ways.

That is not the case for rings. For instance, this answer provides examples of rings that have $4$ elements but are not isomorphic:

$\mathbb{Z}_4,~ \mathbb{F}_2[x]/(x^2+x),~ \mathbb{F}_2[x]/(x^2+x+1),~ \text{and} ~ \mathbb{F}_2[x]/(x^2)$