Do you mean actual divisor corresponding to some rational function? Then look at the zeros and poles (ie. factor the numerator and denominator). Otherwise, as Kevin said divisors are just formal sums of points. They are book-keeping devices that are boring without object that they book-keep for (if this made sense).

I suggest you look at projective curves, where all the nice properties are more obvious. For canonical divisors find divisors of differentials (again the class is canonical, there are many divisors). The differential gives you a transformation between the affine charts (in terms of the original equation of the curve).