Small changes in an external parameter can often lead to dramatic qualitative changes in the Lowest energy quantum mechanical ground state of a correlated electron system. In anisotropic crystals, such as the high-temperature superconductors where electron motion occurs primarily on a two-dimensional square Lattice, the quantum critical point between two such Lowest energy states has nontrivial emergent excitations that control the physics over a significant portion of the phase diagram. Nonzero temperature dynamic properties near quantum critical points are described, using simple theoretical models. Possible quantum phases and transitions in the two-dimensional electron gas on a square Lattice are discussed.