What is motion? I never thought of this before but when an $8^{th}$ standard student asked me to explain what momentum is, I simply said that it the amount of motion contained in a body and tried to explain that how a truck moving at $10\ \mathrm{m\ s^{-1}}$ has more motion than a cat with the same speed (if you don't believe that such a definition even exists then look here ). But he asked that what's the difference between the speed or velocity and motion, if any? I think there must be some slight difference in how we define these. I particularly know of speed and velocity though not motion. (For me it seems that speed rather than being a different quantity is itself a quantifier of motion and hence am a bit jumbled up)
So if possible can someone clarify:


*

*What motion is? 


And in due process contrast it with other quantities such as speed, etc. 
 A: I believe the answer is actually very simple. To be "in motion" simply means to have $\vec{v} \neq 0$. Nothing more, and nothing less. (Thus, as a consequence being in motion implies that speed $\neq 0$)
In physics we don't usually talk about objects having "more" or "less" motion.  It's more of a yes/no thing. I'm not convinced that the truck has "more motion" than the cat.
If you're interested in trying to define momentum for students without resorting to the "it's a conserved quantity so it's useful" argument (I agree that this is often at too high of a level for some introductory physics students), you might want to say something like this: For a bunch of different objects moving at the same velocity, momentum describes how hard it is to alter the velocity of each object. Let's say 10 mph, to the north. Even if you try with all your might you may not be able to change the speed of a train traveling at 10mph. Thus it has large momentum. Meanwhile, you can change the speed of a feather traveling at 10mph with a simple flick of your finger. Thus its momentum is much lower.
A: I don't like these definitions of momentum. They may try to give an intuitive explanation  but some will still find it hard to understand why these definitions correspond specifically to $m\vec{v}$ in Newtonian mechanics.
The truth is that momentum is just a vector that we discovered that it is conserved when there is no external force on the system. It is a helpful definition because it tells you exactly its purpose in physics: to be a quantity that we know that must be conserved, and by taking it into account, we can gain some insight about what happens in systems.
This is important, because in some situations, the momentum isn't $m\vec{v}$ but it is still conserved. For example, to calculate the momentum of a light beam with energy $E$, the correct formula is $|\vec{p}|=\frac{E}{c}$ ($c$ is the speed of light).
One can of course try to find an intuitive explanation for the Newtonian formula, but at the end of the day the answer will always miss a bit the general idea of momentum.
A: 
So if possible can someone clarify:
What motion is?

Momentum is a vector quantity. So "motion" in momentum is velocity $v$ and momentum is $mv$. Velocity is a vector quantity having magnitude and direction. The magnitude of the velocity is often referred to as speed. The speedometer in your car gives you the speed of the car. The speedometer does not tell you what direction the car is going. To know the velocity of the car, and thus its momentum, you need in addition the direction the car is going.
Hope this helps.
A: It's an interesting feature of English that it doesn't have non-technical terms for distinguishing between velocity, momentum and kinetic energy.  In non-technical language "speed" subsumes all three concepts.  German for instance has native words for all three: Geschwindigkeit (velocity), Schwung (momentum) and Wucht (kinetic energy).  Since the concepts are native to the language, people learn the difference between these concepts as kids und understand it intuitively: 


*

*velocity is just how fast something moves between places

*momentum is how much something moves you if you catch it (say, catching a tennis ball vs catching a medicine ball)

*kinetic energy is how much it hurts when you don't catch it (say, getting hit in the chest by a tennis ball vs a medicine ball)


This intuitive understanding can of course also be explained to English speakers, they'll just have to be convinced that these are really different concepts.  On the other hand, calling "momentum" the "amount of motion" doesn't help in developing the instinctive distinction between these three concepts.  It's just using a different word without adding information.
A: I think you want to explain it intuitively, so what I am going to write is how I understand it intuitively. First, I accept 3 things. 
1)An intuitive notion of what POSITION is. 
2)An intuitive notion of what MASS is.
3) I would intuitively expain what FORCE is, a proportion: the harder you push an object, the more force you apply. 
Distance is the relative position between two objects. Speed determines how fast and in what way -is it getting bigger or smaller- that changes. Momentum is mass times velocity and more importantly, it is measures how much force has been applied to the object. To get great momentum you are required to apply great force. 
In this context I would describe MOTION as the process by which we determine the position of the object. Quantity of motion is a physical quantity, usually diferent from the 3 'fundamental' concepts I described above, but directly described by them.
I hope this helps :)
A: "Motion" is a binary quantity -- something is either in motion or not (stationary). (You might want to introduce simple relativity and "frame of reference" here with a simple car example: the driver's coffee cup in the cup holder is stationary in the driver's frame of reference but not in the usual "surface of earth" reference frame.)
"Speed" is motion quantified. Distance unit per time period.
"Velocity" is motion both quantified and with direction included. It might be interesting to note that "direction" here is ambiguous as to 1, 2, or 3 dimensions.
"Momentum" is motivated by the observation that differently massed objects behave differently in collisions with other objects. Heavier objects have "more effect" than lighter-weight objects travelling the same speed. And in fact (Newtonian) physics has determined that as to "constant effect", mass and speed are exactly inversely proportional -- doubling the mass or doubling the speed causes the same change in effect. So it is a useful quantity to multiply mass by speed -- voila Momentum.
Also to note, unlike "speed" vs. "velocity", "momentum" is ambiguous between the scalar and vector (direction included) variety.
Also to note, the "conservation of momentum in closed systems" that (Newtonian) physics has determined. As opposed to, say, velocity.
A: 
What motion is?

Well, it's a bit subjective matter. But to me it is more related with body trajectory in some reference frame
