Density is a measure of how much mass you will find in a given volume. Nowadays we define it as the mass divided by the volume but, interestingly, 'twas not always so. In fact, Isaac Newton (the father of modern mechanics) considered density to be the more fundamental property and defined mass as simply the density multiplied by volume. Sam's question is a very good one: at what scale of the Universe do we start to notice density?
In the nucleus of an atom there are three main forces at work. The electromagnetic force causes protons to repel each other, while the strong force causes them to attract, along with neutrons. The weak force causes protons and neutrons to turn into each other and everything is held in a delicate balance. At this scale all three forces play an equally important role and the result is that all nuclei have the same basic layout.
All atomic nuclei have a fairly equal spacing of atoms meaning that there is an actual "nuclear density" common to all atoms. The value is around 2.3 x 10^17 kg for every cubic meter of nucleus you have.
The space between the nucleus and the electron orbitals also has a pretty consistent density because the distribution of mass is about the same for all atoms. We usually express this in terms of energy per volume rather than mass per volume (because the mass isn't stable) and it has a value of around 1 x 10^-9 Joules per cubic meter. That's quite a lot.
Where density starts to vary is above the atomic scale. Atoms are attracted to each other at a great distance but repelled by each other when they get too close. The sizes of these interactions depend on the size of the atom, the shape it has and how well shielded the nucleus is from the outer radius. Different types of atoms will therefore have different ways of squashing against each other. Ultimately this means different types of atoms will pack together differently and density as the real-world property emerges. So density differences are only relevant on the scale of atoms and bigger. Anything smaller and things are nicely balanced.