If X is an (irreducible) affine algebraic variety, and if U is an open subset of X, then KX(U) will be the fraction field of the ring of regular functions on U.
In fact, the fraction fields of the rings of regular functions on any affine open set will be the same, so we define, for any U, KX(U) to be the common fraction field of any ring of regular functions on any open affine subset of X. Alternatively, one can define the function field in this case to be the local ring of the generic point.
Then it is possible to have zero divisors in the ring of regular functions, and consequently the fraction field no longer exists.
The naive solution is to replace the fraction field by the total quotient ring, that is, to invert every element that is not a zero divisor.
In the particular case of an algebraic curve C, that is, dimension 1, it follows that any two non-constant functions F and G on C satisfy a polynomial equation P(F,G) = 0.