What is a field?

I'm not sure if this answers your question, but a nice line that I'm paraphrasing from an answer from Greg Stevenson over at MO is that if you buy that groups are important for representing symmetries on sets, then you should naturally buy that rings are important for being the things that represent symmetries in abelian groups!

Namely, the set of endomorphisms of an abelian group always forms a ring, and so now if you're looking for automorphisms of an abelian group, you're naturally led to looking at asking question about inverses of elements in rings. The nicest possible world is one in which all of the elements have inverses, i.e., fields.

Of course, this a very narrow response, designed to mirror the question. Fields arise very naturally in very many areas of mathematics (rationals, reals, complex numbers, finite fields, $p$-adics, meromorphic functions, etc.), not all with the intent of modeling the symmetry of an abelian group.

Fields allow us to generalise much of what we take for granted when working over the real, rational and complex numbers: in particular, the operations of addition, subtraction, multiplication and division. Division in particular is what makes a field special, separating it from, say, a ring.

So the short answer to your question is: a field is an algebraic structure on a set which allows us to make sense of addition, subtraction, multiplication and division. These operations are tied together using the underlying group structure and the distributivity law, and what we get turns out to be very useful.

Finite fields, for example, are incredibly useful in cryptography: they allow us to take a finite set of integers and divide one integer by another to get another integer, by using modular arithmetic. There are numerous other important types of fields: function fields, cyclotomic fields, number fields, etc. And furthermore, without fields, we wouldn't have vector spaces!

From a semi-category-theoretic point of view, fields are to commutative rings as simple groups are to groups.

Specifically, a group, $G$, is simple if any only if any homomorphism $G\rightarrow H$ is either a monomorphism or trivial.

In the category of commutative rings, $F$ is a field if and only if any ring homomorphism $F\rightarrow R$ is either a monomorphism or trivial.

Like many things in mathematics, a field is a generalization rather than representation. But, also like many things in mathematics, fields have certain examples that inspired them and form the classical example. The real numbers are the classical example of a field, but certainly not the only one as Complex Numbers, Surreal Numbers, and many others are also prominent examples of fields.

When I first started learning about fields, I thought of all fields as "like real numbers" Obviously that is an oversimplification, but I found it a useful model to get something of an intuitive grasp rather than just trying to deal with naked axioms.

There is one thing that always fascinated me:

Say one day you come across math written in alien symbols, and they all look something like this

$$\blacklozenge \boxtimes\ \blacksquare\ \triangle\ \bigstar.$$

After hours of studying these you come to the conclusion that $\triangle$ denoes ''$=$'' and $\boxtimes$ must be denoting either plus or times. What is it? Well if you find that

$$\blacklozenge \boxtimes \blacktriangledown\ \triangle\ \blacktriangledown$$

and

$$\blacksquare \boxtimes \blacktriangledown\ \triangle\ \blacktriangledown$$

and they don't just use multiple symbols for the same thing, then know that $\boxtimes$ must be times and $\blacktriangledown$ is the zero. Because in addition of the abelian group with ''+'' you always have the unique inverse and the two symbols $\blacklozenge, \blacksquare$ can't both be the neutral element. I think it's pretty cool that these two operations are seperated by such a simple line.

If you have a field and $a$ is an element, then there is also $-a$. From

$$0=b\times a-b\times a=b\times(a-a)\ \ \ \forall b,$$

you can always find this object $0$ in a field by adding any element and its inverse. A field is a set of things you can add with the abelian plus operator like you expect things in the universe to behave if you add them. But there is also this mysterious thing ''$0$'' with

$$0\times a=a\ \ \ \forall a$$

The zero thing. Isn't that object weird?