Why are Nuclei stable and what do neutrons change there? I am currently thinking about nuclear fission and realized that I forgot some basic facts I've learned (and understood... at least I thought so) some years ago in school.
I know that electrons are negatively charged, neutrons are neutral and protons are positively charged. I know that same charged particles repels and different charged particles attract each other. Electrons don't fall in the nucleus, because they are flying around it too fast (at least that is a somewhat reasonable way to think about it - I realize that the details are much more complicated). The nucleus is stable, because besides the electric force there is gravity (which was not so important in that context, if I remember it correctly) and the two nuclear forces. Those make sure that the nucleus is stable.
However, what do neutrons change in this context? Are they behaving similar as protons when it comes to the nuclear forces?
Now, what happens when a neutron hits a 238U?
 A: First, there are some misconceptions in your question: the reason electrons don't fall into the nucleus is not because they're 'flying around it too fast' - it is for quantum mechanical reasons (see here). And it is the strong nuclear force that holds nucleons together (not gravity nor the weak nuclear force).
During fission, the neutron is travelling at a high speed and imparts a large amount of energy (several MeV) into the nucleus that it collides with. This causes the nucleus to break apart into two separate lobes. If those two lobes separate by a great enough distance then, due to the fact that the strong nuclear force acts over a very short range, the electrostatic repulsion suddenly dominates and drives the two lobes apart.
A: The nucleus is held together by the balance of two competing forces. The protons due to their charge repel each other, thus there is no nucleus that is just composed by two protons, because the repulsive electrical force dominates and does not allow a binding.  There exists the deuteron, though, because a proton and a neutron are held together with the strong force, and for the strong force the proton and neutron are indistinguishable and act as a baryon, with an isotopic spin.
This is a basic characteristic of the table of isotopes, and for large numbers of protons many more neutrons are needed to dilute the repulsive force of the charges of the protons.
The whole nucleus can be described by an effective potential which gives energy levels , a shell model  for the available energy locations. This explains the periodic table of elements fairly well, although there are more variations for detail interactions during fission.
