What are the common types of the steam turbines?

condensing turbine and back pressure turbine condensing turbine used a condenser to make the exhaust steam water again and used the same water as boiler feed water to keep it as a closed system.
back pressure turbine's exhaust will be at some pressure, that it can be used for some purposes like distillation, driving a motor etc.
simple cycle turbine and closed cycle turbine simple cycle the exhaust steam is let to atmosphere.
closed cycle turbine, the steam is again condensed into water and used again.
single stage turbine and multistage turbine single stage turbine is small and have only one stage of vane and blade.
multistage turbine are big and have more than one stages of vanes and blades.
impulse turbine and reaction turbine impulse turbine uses the velocity difference of inlet and exhaust steam reaction turbine uses the pressure difference of inlet and exhaust steam.
mostly the industrial turbines are a combination of both impulse and reaction.

in addition to the replies above steam turbines are additionally classified by the following: Steam Pressure: When classified by operating pressure, steam turbines may be supercritical (more than240bar), high pressure (240 to70 bar), intermediate pressure (70 to3 bar), and low pressure (less than3 bar).
Physical Arrangement: When classified by physical arrangement, turbines are either tandem Compound (TC) or cross compound (CC).
Reheat or non reheat Extraction or Non-extraction

There are two types of steam turbines.
First of Non-Condensing (Back-pressure) Steam Turbine and the other is Extraction Steam Turbine .
The primary type of turbine used for central power generation is the condensing turbine.
These power-only utility turbines exhaust directly to condensers that maintain vacuum conditions at the discharge of the turbine.
An array of tubes, cooled by river, lake, or cooling tower water, condenses the steam into (liquid) water.
The cooling water condenses the steam turbine exhaust steam in the condenser creating the condenser vacuum.
As a small amount of air leaks into the system when it is below atmospheric pressure, a relatively small compressor removes noncondensable gases from the condenser.
Non-condensable gases include both air and a small amount of the corrosion byproduct of the water-iron reaction, hydrogen.
The condensing turbine processes result in maximum power and electrical generation efficiency from the steam supply and boiler fuel.
The power output of condensing turbines is sensitive to ambient conditions.