I will try my hand in simplicity...
Momentum is the measure of motion.
So, it is a measure of how much stuff is moving, and how fast that stuff is moving.
What we call mass at the very basic level is "amount of stuff". Very interestingly, more stuff is heavier (pulled more strongly by the Earth - gravitational mass) and harder to move around (change velocity - inertial mass). By some conspiracy of nature (which we think general relativity explains) these are the same to the best of our knowledge. So, more stuff is more stuff, and we call the answer to "how much stuff" the mass.
This all depends on experiment, you see. By experimenting, we have observed that two pieces of stuff moving at a certain speed (let us consider one dimension here, which is sufficient to understand what momentum is) can be exchanged for one piece of stuff (that is, half the mass) which was originally at rest moving double that speed, with the original two pieces of stuff having stopped. You have to arrange the collisions carefully, and avoid affecting the three pieces externally, but this can be done.
So, observation of colliding masses (which is essentially electromagnetic interaction, but let us not digress) with negligible external effects (like friction) you can arrive at the following conclusions:
(1) The amount of stuff moving can change.
(2) The speed of motion can change.
So that is not really a lot. But as it turns out, the product of stuff and speed summed together does not change. This can be confirmed by further experimentation - so we give it a name. Call it momentum. (We could have called it <PERSON>. Not convenient, but possible. A name is a name.)
In trying to describe how this quantity, which is conserved but can be exchanged, is actually exchanged, we introduce the concept of forces. A force is how momentum exchanges hands, so to speak. How much momentum is exchanged depends on how much force is applied and for how much time it is applied. 343
(This even holds relativistically, if you use the relatvisitic momentum.) If the force is internal, that is between two pieces of the system you are studying, one gains momentum while the other loses it -- and in the exact same amount. This is essentially <PERSON>'s third law. But, if you apply an external force (whose reaction force, that is, canceling pair is not within your system) the momentum of the system you are studying will change. You can always expand your system and include more stuff, and ultimately you will have a system whose momentum is conserved. Thus, as far as we believe, the momentum of the universe is fixed.
Although not strictly part of the answer let me add the following:
Conservation of momentum is the coolest conservation law ever. It is always valid. Relativity? Yes (use the correct form). Quantum systems? Yes. Newtonian mechanics? Of course!
Energy is also very carefully conserved by nature, but it is a bit more subtle. When studying macroscopic objects, you need to track heat exchange and internal energy. Also, potential energy is sometimes easy to miss. Momentum? Very nice and simple.
... well, sometimes in charged particle collision, you get a radiated photon which you need to be careful about (it carries some momentum away). But that is significant when things are relativistic. Oh well. 343