The notion of a field (a commutative ring $R$ with $0\neq 1$ and $R^\times=R-\{0\}$) seems to fit uncomfortably into modern algebra. To see what I mean, consider the following statements:
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The concept of a field seems necessary for the foundation of geometry. For example, the "Steinitz Exchange Lemma" (proving that dimension of a vector space is well-defined) seems to require a field. Dimension of a vector space, in turn, seems to underlie all other notions of dimension in mathematics.
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The category of fields is not much of a category. It doesn't have products or coproducts. Every morphism is a monomorphism. It's not connected (there is one connected component for each characteristic). Free fields don't exist. In general, fields do not form an "equational theory" so they don't behave like an "algebraic category".
So what is a field, really? Here are a couple suggestions to get you started:
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A single field $K$ is something like the category of sets. This is suggested by the analogy between the bifunctor $\mathrm{Hom}_{\mathcal{C}}(-,-):\mathcal{C}^{\mathrm{op}}\times\mathcal{C}\to\mathsf{Set}$ on a category $\mathcal{C}$ and a bilinear form $\langle -,-\rangle:V\times V\to K$ on a $K$-vector space $V$. It also agrees with the use of the "ground field" in algebraic geometry, which is something like the "universe of discourse".
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The construction of the splitting field doesn't have an UMP because fields have too many automorphisms. So it seems that "Galois theory" is what makes fields special. Perhaps a formalization of Galois theory can help explain what fields really are.
Best Answer
Fields are the simple (no nontrivial quotients) commutative rings. Grothendieck told us to work in nice categories with nasty objects rather than nasty categories with nice objects; fields are the nice objects, and the nice category they live in is all commutative rings. You want to work in all commutative rings and sometimes study fields the same way you want to work in all modules and sometimes study simple modules.