The DiVincenzo criteria, a list of necessities for the construction of a quantum computer is summarized herein. In reviewing the physics underpinning the trapped ion qubit paradigm, I introduce a rotor model for atoms/ions and demonstrate how single qubit gates, such as the phase and Hadamard gates, are realized in this framework. We use the latter to illustrate how ions respond to laser radiation, the mechanism by which ion qubits are addressed. We show how the Cirac-Zoller mechanism, and its generalization, enables the realization of two-qubit control gates. I provide a survey of trapped ion qubit systems, including optical and hyperfine qubit systems.