We introduce a statistical mechanical model for the physical properties of water. Each water molecule is a two-dimensional disk with three radial hydrogen-bonding arms. Energetic interactions are based on water triplets, a central water molecule interacting with two neighbors via hydrogen bonds, van der Waals attractions, and steric repulsions. Interactions with more distant molecules are treated in a mean-field way. Each molecular triplet can be in one of three energy levels: cage-like hydrogen-bonded structures, denser nonbonded structures, and expanded structures with no near-neighbor interactions. The model predicts water's thermodynamic anomalies, including maxima in density, minima in isothermal compressibility and heat capacity, and expansion upon freezing at low pressure. It predicts the main features of water's phase diagram, including multiple crystalline phases and the proposed liquid-liquid transition in supercooled water. Also, it captures qualitatively the fragile-to-strong transition in the liquid's temperature-dependent relaxation processes. This model is intended to give simple insights into the microscopic origins of water's distinctive physical properties.