Definition of momentum We say that momentum is the measure of how a body is moving or the quantity of movement inside a body
But what this definition really mean?
This terms are very vague
$p=mv$,why the movement inside the body depend on it's mass?
 A: (In classical mechanics), the definition of momentum is $\vec{p}=m\vec{v}$.
The reason this is a good definition is because it is useful. In particular, the momentum of a collection of particles that are not in an external potential is conserved. Conserved quantities make it possible to understand aspects of the behavior of a system without solving complicated equations.
My advice would be not to get stuck on any philosophical musings on "why this definition." You can make any definition you want; the reason definitions stick around and make it into textbooks is because they are useful and help us solve problems.
The above reasoning is good enough reason to justify the definition of momentum -- we make a classical mechanics definition, and we see a benefit of using that definition to solve classical mechanics problems. But in fact, momentum is in some sense even better than it needs to be. In particular, when we generalize classical mechanics by (a) introducing relativity, and (b) moving to quantum mechanics, we find that many concepts like force or velocity do not have nice translations into those more general frameworks, but momentum does.
A: The definition is simply $\mathbf{p} = m \mathbf{v}$.
In classical mechanics, this quantity is conserved in absence of an external net force, since the second principle of dynamics reads
$\dfrac{d \mathbf{p}}{dt} = \mathbf{F}^{ext}$,
and thus $\mathbf{p}(t) = \text{const.}$ if  $\mathbf{F}^{ext} = \mathbf{0}$.
In order to understand the meaning of the mass in the momentum, you need to evaluate the influence a force (and the impulse of the force, i.e. its integral in time) in changing the momentum
$\displaystyle m (\mathbf{v}_1 -  \mathbf{v}_0) = \mathbf{p}_1 - \mathbf{p}_0 = \int_{t_0}^{t_1} \dfrac{d \mathbf{p}}{dt} dt = \int_{t_0}^{t_1} \mathbf{F}^{ext}(t) dt = I$.
A: An object's momentum is the product of its mass and its velocity.
$$\vec P = m \vec v$$
Nothing vague about it.
Why is this referred to as the "quantity of motion"?  That's more of a history os science question.
A: There are two sorts of answers you could go for.
First, remember that forces are defined as something that makes a motion not linear and uniform for a point-like system. The next question is, if you apply a given force to a variety of systems, do they all react the same, i.e. do their respective motions are modified identically? If that were true, then Newton's second law would be:
$$\frac{d\vec{v}}{dt}=\vec{F}$$
But of course this doesn't match reality. Heavier system are less affected. This means that mass has a role to play in the way velocity is affected by forces.
Trials and errors led to the conclusion that the useful quantity inside the derivative had to be the product $m\vec{v}$, at least in simple cases (when there isn't any electromagnetic field).
This is especially useful when studying collisions. If you throw a ball with a given velocity against a wall, not much will happen. But if you throw a truck with the same velocity, the wall will likely be destroyed. So collisions aren't just a matter of velocity. Again, the product $m\vec{v}$ proved to be the appropriate quantity, i.e. the one that is conserved during the collision.
Finally, a more modern answer is: $m\vec{v}$ is the conserved quantity that appears due to the fact that the laws of physics are the same at every point (one use of Noether's theorem).
A: Simple answer:
Usually, you consider the time that you're applying a force. Some combination of force and time is basically my idea of "strength". That combination should allow variation in force over time, so it should be an integral. This integral formula gives $Ft$ for constant force.   Because $Ft$ is $\Delta mv,$ this momentum is basically how much strength you need to stop an object. The "quantity of motion" is basically proportional to the amount of strength required to stop the object, since I'm pretty sure nobody would set speed limits for something as light as a tennis ball because they aren't that dangerous.
