Some of the methods of classification of turbines are
According to position and mounting
Vertical Turbine
Horizontal Turbine
According to the head of water
Low Head Turbines
Medium Head Turbines
High Head Turbines
Eponymous classification (According to the name of the originator)
Francis Turbine
Kaplan Turbine
Pelton Wheel
Girard Turbine
According to the Flow Direction
Radially outward flow turbine
Radially inward flow turbine
Axial Flow Turbine
Mixed Flow turbine
According to Discharge
Low discharge
Medium discharge
High Discharge