Hmm...everything?

https://en.m.wikipedia.org/wiki/Field_(mathematics)

IANAM but for me the significance and meaning is to do with being able to reason about problems at a higher level of abstraction.

E.g. the proof using Galois Field Theory that the heptagon is unconstructable:

https://en.m.wikipedia.org/wiki/Constructible_polygon

Aside from the obvious fact that the set everyday numbers (both rationals and reals) are fields, most crypto heavily relies on finite fields.

[edit] and come to think of it, I might be mistaken I don't think the notion of vector space and therefore the whole of linear algebra would work without a field for the underlying coordinates.