(This is not meant to be too rigorous, so please bear with me)
For example, if you add, subtract, or multiply two integers, you get another integer. So the integers are a ring.
On the other hand, the rational numbers are closed under division too, which makes them a field.
So it's now interesting to see which ones can be divided by which!
So rings are not just numbers, functions can also form a ring. You can add, subtract, and multiply them.
Or, put another way, you know the ring, you know the shape.
This is one motivation for the subject of algebraic geometry.
See, ideals were originally invented for number theory. But again, rings are not just numbers!
You know the ring, you know the shape.
They are a ring after all, and determine a shape.
You know the ring, you know the shape. At least we should...