Are there other Density Functional Theories other than Kohn Sham DFT?
KS-DFT is the basis for all orbital-based DFT Formulations. In it you have different specialities (constricted DFT, TD-DFT, rs-DFT to name a few). 

KS-DFT is used for molecular problems, because of atom bonds etc.

TD-DFT is used for time-dependend problems like excitation energies, polarizabilities etc.

RS-DFT is using range-separated functionals (long-range, short-range) to handle effects better in the long-range domain.

Addition by @pentavalentcarbon:
Current cDFT implementations are based on KS-DFT and perform a real-space partitioning of the electron density.

Plane Wave DFT (no orbitals, electron density calculated with plane waves, was the first DFT) is used for periodic problems like metals etc.

DFTB (Density Functional Tight Binding) is good for Molecular Dynamics since its an approximate DFT and allows handling of larger systems and greater timescales.
Addition by @pentavalentcarbon:
A good analogy for DFTB would be that it is similar to semiempirical methods (PM7, other NDDO parametrizations) where certain integral matrix elements are replaced with precomputed, fitted parameters, but there is a minimal basis set present.

KS-DFT is dominant because as computational chemists we are mostly working with molecules. As that we need a simulation of atom bonds. There is a good Book about DFT which I can only recommend (A chemists guide to Density Functional Theory): http://www.goodreads.com/book/show/70706.A_Chemist_s_Guide_to_Density_Functional_Theory