Laser-cooled 9Be+ ions confined in two-dimensionally extended lattice planes were directly observed, and the images were used to characterize the structural phases of the ions. Five different stable crystalline phases were observed, and the energetically favored structure could be sensitively tuned by changing the areal density of the confined ions. The experimental results are in good agreement with theoretical predictions for the planar (infinite in two dimensions) one-component plasma. Qualitatively similar structural phase transitions occur, or are predicted to occur, in other experimentally realizable planar systems.