Two-dimensional chromium ditelluride (${\mathrm{CrTe}}_{2}$) is a promising ferromagnetic layered material that exhibits long-range ferromagnetic ordering in the monolayer limit. The formation energies of the different possible structural phases (1T, 1H, 2H) calculated from density functional theory (DFT) show that the 1T phase is the ground state, and the energetic transition barriers between the phases, calculated by the nudged elastic band method, are large, on the order of 0.5 eV. The self-consistent Hubbard $U$ correction parameters are calculated for all the phases of ${\mathrm{CrTe}}_{2}$. The calculated magnetic moment of 1T-${\mathrm{CrTe}}_{2}$ with $\ensuremath{\ge}2$ layers lies in the plane, whereas the magnetic moment of a monolayer is out-of-plane. Band filling and tensile biaxial strain cause the magnetic moment of a monolayer to switch from out-of-plane to in-plane, and compressive biaxial strain in a bilayer causes the magnetic moment to switch from in-plane to out-of-plane. The magnetic anisotropy is shown to originate from the large spin-orbit coupling (SOC) of the Te atoms and the anisotropy of the exchange coupling constants ${J}_{xy}$ and ${J}_{z}$ in an XXZ type Hamiltonian. Renormalized spin wave theory using experimental values for the magnetic anisotropy energy and Curie temperatures provides a range of values for the nearest neighbor exchange coupling.