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.