All fits with the presence of a subatomic particle without charge, of a neutral nature considering electric charge. It's located in the nucleus at the atom's center, or at least this is the current theory. While this field is heavily in experimentation, the reality of this particle cannot be doubted as it is the main cause of well-known processes, like nuclear chain reactions or neutron activation.
When an unstable atom is attacked with neutrons, there's a quantifiable chance to nuclear fission. If that happens, another number of neutrons are going to be released, allowing a set of chain reactions. Fusion also releases neutrons in the exothermic reaction. This is how Nature wants it, so sooner or later someone is going to discover it, using the energy for different purposes, always creating an extreme risk in that period. We hope until it's completely forbidden by rational means.
Sometimes the free propagation of neutrons could induce changes in other atoms, creating for instance radioactive isotopes. This is a part of neutron activation.
The neutron right now looks to be the union of three quarks thanks to other particles —well, perhaps not strictly particles— called gluons. Fortunately, these have generic names because it could be a bit grandiloquent to read: "The Universe is formed of John Doe or thanks to the John Doe mechanism". Perhaps a better word instead of grandiloquent would be absurd, but funny in context because it's not the first time!
How many neutrons in an atom? It depends, the periodic table is meant to provide these info. It's done subtracting the proton number from the atomic weight, after rounding the quantities.