The density of an object is how much the space bound by the object is filled with matter (atoms). The closer the atoms or molecules in the object are to each other, the less “nothing” there is and the more dense the object. A box of air isn’t very dense because air is a collection of gas molecules that are relatively far apart with nothing in between. A rock is more dense than air because the atoms that make up the minerals in the rock are all bound closely together.
There are two ways to define mass. The material definition of mass is a measure of the total amount of matter in an object (where density is a per volume measure), so this is the density times the volume1 \(M=\rho V\), where \(\rho\) (the greek letter rho) is density. The inertial definition is, mass is the property of an object that resists acceleration (this property is inertia). To understand this, we need to know what makes an object accelerate, which is a force.
A force is the something applied to an object that potentially causes the object to accelerate. A force isn’t necessary for an object to move. A force applied to an object slows it down or speeds it up. So blood moving through an artery is slowed down by friction (a type of force) and speeded up by the heart pressurizing the blood (another kind of force).
Newton’s second law states that force is the product of mass times acceleration2 \(F=MA\). We can re-arrange this to \(A=\frac{F}{M}\). Given the same force applied to two objects, the more massive object (bigger \(M\)) will have a smaller acceleration. So this is the inertial definition of mass: mass is the property of an object that resists acceleration. This concept leads directly to the concept of momentum.
Momentum is the mass of an object times its velocity3 \(p = M\nu\), where \(\nu\) is the greek letter nu. We usually think of momentum as we would inertia: an object with more momentum resists change in direction and/or speed more than an object with less momentum. But it’s really the mass (inertial) component of momentum that makes this so.
Finally, note that the change in momentum over time is \(\frac{\Delta M \nu}{T} = M \frac{\Delta \nu}{T} = MA = F\)! That is, force is the change in momentum over time.