We obtain a phase diagram of an anisotropic antiferromagnetic $XY$-model of classical Heisenberg spins on a triangular lattice by calculating the average values of $x$-component and $y$-component of spin, the chirality, angles between averaged spins on sublattices and the specific heat by means of Monte-Carlo simulations. It turn out that the system has two phase transition temperatures: one for the Kosterlitz-Thouless-type phase transition and the other for the Ising-type phase transition. These two phase transition temperatures are merged into one in the isotropic $XY$ model. We find that the chirality is a good order parameter at low temperatures even in the system with anisotropic interaction. For the system with isotropic interaction, the 120 degree structure persists as far as the chirality is non-zero.[pdf]