EDIT: The comic says the strong force holds protons and neutrons together. I know the strong force is why the nucleus doesn't fly apart because of the electromagnetic force, but I thought it was more a force between two protons, or is there also an interaction between neutrons and protons? I suppose there must be, otherwise neutrons would just sort of float around...
The strong interaction is a force between quarks and gluons, mediated by gluons. Quarks are the thingies that make up protons (quarks up, up, down, and a soup of temporary particles) and neutrons (quarks up, down, down, and a soup of temporary particles). Protons, neutrons, and other exotic particles made of quarks exist because the strong force binds their components together.
Atoms are similar, only more complicated, because it's the "residual" part of the strong force that binds protons and neutrons together; some models of atomic interactions dispense with the complex strong force altogether, and modelize the "nuclear force" as protons and neutrons exchanging pions (the lightest particle made of quarks) with each other. As far as the strong force is concerned, there's not much difference between neutrons and protons, they are just "bags" of quarks.
So, neutron-neutron, proton-proton and neutron-proton can interact through the strong force.
Addendum: the electromagnetic interaction is not the only mechanism that tries to split atoms apart. The other is Pauli repulsion (it also goes by other names). The basic concept is that two fermions can't be in the same quantum state, so if you want two protons (resp. neutrons) in an atom, they must have different energies. So as an atom becomes larger, the extra protons (resp. neutrons) in the atom must have increasingly large energies. And because matter tries to minimize its energy, that means that the extra protons (resp. neutrons) will try to leave the atom. The strong force is what keeps them together.