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 non-trivial emergent excitations which control the physics over a significant portion of the phase diagram. Non-zero temperature dynamic properties near quantum critical points are described using simple theoretical models. Possible quantum phases and transitions of the two-dimensional electron gas on a square lattice are discussed, including phases with spin-charge separation and their experimental signatures.