How do we define what is "External" force or "Internal" force in the context of momentum conservation? I know that without presence of any "External" force momentum is always conserved. But how do we distinguish between "External" force and "Internal" force where all are "Force"?
 A: You define a system which you are interested in.
If there is no net external force acting on the system then linear momentum is conserved.
You can identify internal forces as they must occur in equal in magnitude but opposite in direction pairs - Newton's third law.
So you find a force in the system $\:\mathbf{f}_{12}\:$ which is the force on part $1$ of the system due to part $2$ of the system which has its equal in magnitude opposite in direction twin, $\:\mathbf{f}_{21}\:$ force on part $2$ of the system due to part $1$ of the system.
There is no such pairing of forces within the system for external forces which are forces on the system due to something outside the system so their Newton's third law pair would be a force on something outside the system due to force produced by system.

A: For physical systems  the external forces include 
1.  Applied Forces
2.  Normal force of reaction
3.  Force due to strings(tension)
4.  Frictional forces between surfaces in contact
    Etc.
Whereas internal force include
1.Force due to gravity between masses
2.Magnetic force between magnetic bodies
3.Electrical force 
4 spring force  etc.
To identify force as internal and external one must define the system and its surrounding and then go for classification.
If parts of the same system exert forces on each other ,then they form pairs of internal forces
Internal forces operate such that the overall mechanical energy of a system does not change .
If we have a mechanical system the external forces  are those which applies between part of a system and its surrounding/environment.
Some illustrations:
A batsman hits a cricket ball- the particles of the ball holding together are due to internal forces but the bat forcing the ball through a hit is external force on the ball.
If motion of a lift is our system then a person jumping or playing a ball, doing some exercises inside the lift is due to internal force only .
A vehicle is moving and parts of the vehicle are applying internal forces but the force of friction being applied by the road is external force on the vehicle.
similarly the push of the vehicle   on surface of the road and the normal reaction on it are external force .
It is a misnomer to look for internal force using Newton’s third law of action and reaction . As the pair of these forces act on different bodies and are external to them individually.
