I'll offer another "explanation" for rings:
a ring (see here) is a
monoid in the monoidal category of abelian groups (with respect to the standard tensor product of abelian groups).
This perspective is useful in that it shows what the right generalizations and categorifications of rings are. This is a general phenomenon: when you want to know which of several equivalent definitions is the fundamental or right one, check for which of these you can find natural oo-categorical versions. The more natural a concept, the easier it generalizes this way.
For rings, we notice that the category of abelian groups is the archetypical abelian category. The oo-version of an abelian category is a stable (oo,1)-category. The archetypical one is the (oo,1)-category of spectra - Spec. A commutative monoid in Spec is an "commutative oo-ring" usually called an E-oo ring. If it is non-commutative it is called an A-oo ring.
This is the story about rings and their vertical categorifciation. There is also insight into the nature of rings to be gained from their horizontal categorification:
a monoid in Ab, hence a ring, is equivalently an an enriched category with a single object over the category of abelian groups:
write pt for the single object of an Ab-enriched category, then Hom(pt,pt) is an abelian group equipped with a homomorphism of abelian groups Hom(pt,pt)otimesHom(pt,pt) --> Hom(pt,pt) that is associative and unital. So Hom(pt,pt) is some ring, and every ring works.
So a general Ab-enriched category may be thought of as a ringoid, if you wish.