When is momentum not conserved? What are some common examples where momentum is not conserved?
This question arose in my mind when I read that a ball dropped from a height penetrates into a bed of sand and that momentum is conserved.
 A: If you have a system $A$ with non-conserved momentum ${\bf p}_A$,
$$
\frac{d}{dt}{\bf p}_A = {\bf F} \ne 0,
$$
then you have simply chosen not to consider the larger system $A+B$ with conserved momentum ${\bf p}_A + {\bf p}_B$,
$$
\frac{d}{dt}\left({\bf p}_A + {\bf p}_B\right) = 0,
$$
where
$$
\frac{d}{dt} {\bf p}_B = -{\bf F}.
$$
For example, if a ball (system $A$ above) dropped from a height penetrates a bed of sand on the Earth (system $B$ above) and comes to rest, so that the ball has a change in momentum $\Delta p$ in the direction opposite the ball's motion, then the Earth* has a change in momentum $\Delta p$ in the direction of the ball's motion, so that the total change in momentum is 0.
*Technically, the Earth plus everything else in the Universe.
A: As lemon says, momentum is always conserved.
Nevertheless, there are situations where we do not want to take the full system into account. We instead write an effective description in terms of a reduced set of variables. If the neglected degrees of freedom can absorb momentum, then the effective theory for the interesting variables looks like it does not conserve momentum.
For example, a coin sliding on a table experiences a friction force. If you give it some speed and let it go it spontaneously stops. In the theory that takes only the coin into account, the momentum is not conserved. Of course the momentum hasn't disappeared. It went in imperceptible movement of the table, the ground, etc which were neglected.
Fundamentally, momentum conservation is linked to invariance under space translations. See Noether's theorem. If you want to find a system that does not conserve momentum you should look for situations where space is not uniform, e.g. balls rolling on the surface of a bowl, a planetary system (when the dynamics of the sun is neglected), etc.
A: Momentum is not conserved if there is friction, gravity, or net force (net force just means the total amount of force). What it means is that if you act on an object, its momentum will change. This should be obvious, since you are adding to or taking away from the object's velocity and therefore changing its momentum. 
Bonus: Momentum actually is conserved when gravity acts on an object, because for gravity to exist, it must be between two objects, and the objects experience equal and opposite force.
A: Momentum is conserved in case of motion under gravity because a object is pulled by the earth and the earth of pulled by the object.
The momentum terms of the object and earth cancel each other out as the object has small mass but high velocity and the earth has very high mass and very small velocity in the opposite direction of the velocity of the object.
Internal forces cannot change the total mechanical energy but can only transform them. Gravity is an internal force, so it cannot change the net mechanical energy but it can transform KE into PE and PE into KE. 
A: Momentum conservation is defined for a system. For example in case of explosion of a particle into many fragments momentum is conserved in the x direction but not in the y direction where gravity is acting but if we consider the particle and Earth as a system the gravity force becomes an internal force and hence momentum is conserved for that system.
So in the former system gravity is a external force and in latter gravity becomes an internal force. Momentum conservation take place only when no external force acts on the system. (Note that the system we are considering may be experiencing external forces all around from various parts of the universe. However from the reference frame fixed to us on the Earth we don't notice the effect of such forces and therefore not taken into account. Understand it in this way. You are in a long cart  which is accelerating horizontally. You throw a ball with velocity u horizontally. It breaks into small pieces. In the reference frame fixed to you you can apply momentum conservation in the x direction because no external force is acting in that frame. But in the reference frame fixed to a person standing on the road momentum conservation in the x direction doesn't apply because he sees an external force acting on ball in x direction. So indeed momentum conservation depends on frame of reference too.)
However for practical purposes we often apply momentum conservation equation in the direction of gravity when the impulsive forces are larger than gravitational force so that it  is ignored.
A: in case of a photon striking to a mirror..momentum is not conserved
                  P+O=-P+Pm not valid as:
              2P#Pm(2P cant be equal to Pm)

