We all know that atoms are comprised of electrons and a nucleus. The nucleus is tiny and dense with positive protons and neutral neutrons, while the electrons orbit far away and are negative. So then, why don't atoms fall through other atoms if there's so much empty space in between? Two reasons really: the electromagnetic repulsion and the Pauli exclusion principle.
The first is simple. When you bring like charges together they repel, and this force is proportional to the inverse of the distance between the two charges squared. This means that if you bring objects closer and closer together, the resisting force will become greater and greater until it overcomes the force pushing the two objects together.
The second theory could kill you if you aren't careful, so take breathers in the middle. Quantum mechanics dictates that electrons are in every possibility at once. So, really, there is no empty space between the electrons and the nucleus: it's all filled with possibilities. However, Pauli's exclusion principle also dictates that no two identical fermions (let's just say this includes electrons and move on) may occupy the same quantum state simultaneously. Thus, because both the electrons from one atom and the electrons from another atom cannot exist in the same place, but still fill up their surroundings with possibilities, at a certain point they become incredibly hard to push into each other any further. At this point you have what's called degenerate matter.
Thus, crushing atoms into each other is almost impossible. The only reason we can pass through liquids and gasses is because we simply push them into the surrounding empty space or around each other.