(i) External combustion engine:

An engine in which combustion of fuel takes place outside the engine cylinder is called external combustion engine. These engines are generally called EC engines. 
Ex: Steam engines, steam turbines, closed cycle gas turbine etc. 

(ii) Internal combustion engine:

An engine in which combustion of fuel takes place inside the engine cylinder is called internal combustion engine. These engines are generally called IC engines.
 Ex: Petrol engine, diesel engine, gas engine etc.

The following are the most important ways of classification of I.C. engines:

(i) According to the type of fuel used 

(a) Petrol engines: In this type of engines, the fuel used is petrol. 
(b) Diesel engines: In this type of engines, the fuel used is diesel. 
(c) Gas engines: In this type of engines, the gaseous fuels like natural gas, biogas, are used.
(d) Bi-fuel engines: These engines use a mixture of two fuels.

(ii) According to the number of strokes per cycle 

(a) 4-stroke engine: In this type of engines, the working cycle is completed in four different strokes.

(b) 2-stroke engine: In this type of engines, the working cycle is completed in two different strokes. 

(iii) According to the method of ignition

(a) Spark ignition engine (S.I.Engine):In this type of engines, fuel is ignited by an electric spark generated by a spark plug. 
(b) Compression ignition engine (C.I. Engine): In this type of engines, the fuel gets ignited as it comes in contact with the hot compressed air. 

(iv) According to the cycle of combustion

(a) Otto cycle engine: In this type of engines, combustion of fuel takes place at constant volume 
(b) Diesel cycle engine:In this type of engines, combustion of fuel takes place at constant pressure

(c) Duel combustion engine: In this type of engines, combustion of fuel first takes place at constant volume and then at constant pressure

(v) According to the number of cylinders 

(a) Single cylinder engine:This type of engines consists of only one cylinder. 
(b) Multi cylinder engine:This type of engines consists of 2, 3, 4, 6 or 8 cylinders

(vi) According to the arrangement of cylinders 

(a) Vertical engine:In this type of engines, the cylinder is arranged in a vertical position 
(b) Horizontal engine:In this type of engines, cylinder is arranged in horizontal position.
(c) Inline engine:In this type of engines, cylinders are arranged in line
 (d) Radial engine:In this type of engines, cylinders are arranged along the circumference of a circle. (e) V-engine: In this type of engines, combination of two inline engines equally set an angle.

(vii) According to the method of cooling

(a) Air cooled engine:In this type of engines, the heated cylinder walls are cooled by continuous flow of air 
(b) Water cooled engine: In this type of engines, water is used for cooling the heated cylinder walls.