We study the distribution of well-rounded sublattices of Z^2 by means of investigating the structure of the set C of its similarity classes. We prove that C has structure of an infinitely generated non-commutative monoid, and define the notion of minima and determinant weight for each similarity class in C. We show that these similarity classes are in bijective correspondence with certain ideals in Gaussian integers, and construct an explicit parametrization of lattices in each such similarity class by elements in the corresponding ideal. We use this parametrization to investigate some basic analytic properties of zeta function of well-rounded sublattices of Z^2, including its order of the pole, growth of coefficients, and some related features.