We report a Monte Carlo study of phase transitions in a finite two-dimensional (2D) system of charged classical particles which are confined by a circular parabolic or hard-wall well. The ground-state configurations are found by static energy calculations and their structures are analyzed using the Voronoi constructions. A Mendeleev table for these classical 2D-like atoms is obtained. We calculate the radial and angular components of the displacements of the particles as functions of temperature and determine the critical temperatures for the order-disorder phase transitions. The intershell rotation and intershell diffusion transitions are investigated. The results are compared with Wigner crystallization in the infinite 2D system.