There are two forces involved in holding atoms together. These are the electromagnetic force and the strong nuclear force (there is also a weak nuclear force that is involved with radioactive decay, but it's a side-show).

The electromagnetic force causes the negatively charged electrons to be attracted to the positively charged protons; this is the force that holds the electrons onto atoms. Neutrons, as you observe, have no electric charge, and so the electromagnetic force has no influence on them. They are glued to the protons via the strong nuclear force, as they do possess properties of the nuclear forces.

I do not understand the nuclear forces well enough to be able to explain why atomic nuclei require the numbers of protons and neutrons that they do, or why some isotopes are stable and others are not.

All elements have many isotopes; those with seemingly fewer have fewer *stable* isotopes. I observe that generally the ratio of protons to neutrons decreases as the atomic number goes up; for instance, hydrogen has one proton and zero or one neutrons for its stable isotopes, carbon has six protons and six or seven neutrons, and calcium has twenty protons and twenty-something neutrons. I do not know why.

Not all atoms need neutrons --- the hydrogen atom has no neutrons. As the atomic number (the number of protons) increases, the number of neutrons also increases. This is because protons have positive charges and the only reason that they are willing to stay together in the nucleus is because the neutrons effectively act as "filters" for these positive charges by interacting with the protons themselves.

Isotopes are found because for a given number of protons in the nucleus (meaning for a given element) the precise number of neutrons required to make the element stable can vary. While one particular number of neutrons might correspond to greatest stability, other neutron counts may also be acceptable.

I'm guessing from your question that you already know atoms are made up of three things: protons (which determine the element, or the atom's properties), electrons (which determine the atom's reactivity, to some extent) and neutrons (which determine the atom's isotope). I don't know if it's necessarily true that an atom "needs" any of these things. That's just what atoms are made up of.

You are correct that certain atoms have lots of isotopes (e.g. carbon, xenon) while some have only three (hydrogen). Certain atoms have radioactive (unstable) isotopes while other atoms do not. I've never asked myself why this is the case. That's a really good question. It probably has to do with quantum mechanics, but the truth is, I really don't know.