Ring

In the context of algebraic geometry, a ring is a set $$R$$ equipped with two binary operations $$+$$ and $$\cdot$$, called addition and multiplication, such that:

The additive and multiplicative identities are denoted $$0$$ and $$1$$, respectively.
 * $$R$$ is an abelian group under addition;
 * $$R$$ is a commutative monoid under multiplication; and
 * Multiplication distributes over addition, i.e. $$a \cdot (b + c) = a \cdot b + a \cdot c$$ for all $$a, b, c \in R$$.

Note that this is what might be called a "commutative ring with unity" in a more general context.

Etymology
The name apparently comes from the early days of algebraic number theory, when rings of cyclotomic integers were being studied. These contain idempotent elements, namely the roots of unity, and such an element can be said to "circle back around" to itself when multiplied by itself repeatedly. This is the origin of the term "ring."

Given that not all rings exhibit this behavior, and that even in those that do it's not the fundamental thing of interest, the name is arguably not a very good one.

Motivation
Two classes of examples motivated the development of ring theory: on the one hand there are the rings of integers in number fields, and on the other there are the rings of coordinate functions on geometric objects. Observed similarities between these two classes of objects were major motivations for the development of commutative algebra and modern algebraic geometry.