The "typical" pheasant genus Phasianus in the family Phasianidae consists of two species. 

1/ Common pheasant (Trĩ đỏ, Phasianus colchicus), also Ring-necked pheasant

Galliformes is the scientific (Latin-based) name for the order of heavy-bodied, ground-feeding birds like chickens, turkeys, pheasants, and quail; the name itself comes from the Latin word "gallus," meaning "cock" or "rooster". 

The family name Phasianidae comes from Latin phasianus, meaning "pheasant," which itself derives from the Ancient Greek word phāsiānos (φᾱσιανός), meaning "bird of the Phasis," named after the ancient river Phasis (now Rioni) in Georgia, where these birds were known to Europeans. So, the root is Greek (Phasis) -> Latin (Phasianus) -> New Latin (Phasianidae). 

The genus name is Latin for pheasant.