Ion-gated ion channels
The main families of calcium-gated channels I am aware of are the calcium-gated potassium channels and the internal ryanodine receptors involved in calcium-induced calcium release - you could start from those linked Wikipedia articles.

I wouldn't really make these a fourth category, they should either be thought of as voltage-gated channels that are modulated by intracellular calcium, or as ligand-gated channels - the ligand just happens to be a calcium ion.

Some channels in this category are not influenced by calcium directly but via calmodulin, which is a common mediator of calcium-driven intracellular influences.