Momentum and relative motion To calculate momentum, $p=mv$, you must know the value of $v$. However, since all motion is relative, there is no abslute value for $v$. So,  what is it you are actually calculating? It seems like the answer depends on the frame of reference.
 A: Yes, momentum does change from one reference frame to another. 
In Newtonian mechanics it is a 3 vector. Also velocity. 
In special relativity it is part of a 4 vector, with the fourth component (or more often the zeroth) the energy. It is the energy momentum vector.  It transforms in Minkowski spacetime as a 4 vector. In general relativity is is part of a 4 by 4 energy momentum stress tensor. 
So it means something in the reference frame you are observing it as. If you change reference frame, say you start moving in some direction at constant speed, the components of the velocity and momentum change. In Newtonian mechanics like a Galilean transformation (the simplest example is in 1 dimension, where you might see 3 m/sec, and if I'm moving at 1 m/sec with it I'll see 2 m/sec), in special relativity there Is a Lorentz transformation that changes the whole 4 vector, and with it the velocity vector components, and the momentum. In general relativity a little more complex but the same idea. 
Velocity and momentum are always dependent on the observers reference frame. 
A: You are correct that the momentum of an object depends on the frame of reference of the observer.  However, in a collision, the change in momentum of each object in the collision is found to be the same by all observers in all inertial frames of reference.
A: You are absolutely right; the value of the momentum of an object depends on the reference frame in which it is measured. The interesting question that might be bothering you about this is then: 
How can momentum possibly be useful if its value can be anything?
Am I right? If so, here is the simple answer: 
The laws of physics pertaining to momentum are true in all (inertial) reference frames, regardless of the specific value of momentum in any particular frame.
Most importantly, the law of Conservation of Momentum only requires that the total momentum of a closed system remain constant in time. One frame might measure the total momentum to be 30 kgm/s in some direction, while another might measure it to be 1500 kgm/s in another direction, but provided you don't change reference frames throughout your observations, your measured value will stay constant in time.
Similarly, Newton's Second Law can be expressed as $F=\frac {d\vec p}{dt}$, for some force-law $F$. Again, the law only involves the rate of change of momentum, $\frac {d\vec p}{dt}$, and not the value of $\vec p$ itself. And all inertial observers will measure the same value for this rate of change.
An interesting law that at first does seem to depend on the actual value of momentum occurs in relativity, which asserts the invariance of the mass $m$ of an object, as defined (in units where $c=1$) by:$$m^2=E^2-|\vec p|^2$$
It turns out though that the energy $E$ of a body is also frame-dependent, and when you do the subtraction you get a number which is the same in every frame!
In short: 
The value of momentum depends on the reference frame, but the laws involving momentum do not.
