Neutrons act as a kind of "glue" that holds nuclei together. Remember that the nucleus of an atom is positive (once you get "inside" the orbits of the electrons, their negative charge has no effect), and positive charges repel each other, as do negative charges. So if you're going to pack lots of small positive charges together (the protons), that may work for small nuclei, but when you get to larger and larger nuclei you'll find it harder and harder to hold them together, without some other kind of help. The "help" is the neutrons - particles with no charge, but with the same "strong nuclear force" (one of the four basic forces of nature) as protons. Imagine trying to hold together a crowd of people angry with each other - you'll have more luck if you intersperse some "neutral" people whose job it is to keep the angry people separated a bit, but not so separated that they can't talk to each other.
You get to a point where, for large nuclei like that of uranium, you need many more neutrons than protons to get the job done. That's why, with nuclear fission (where you split a uranium atom in half), you get a few neutrons flying away from the reaction - they aren't needed anymore with the smaller nuclei that are left over.
The reason atoms have different numbers of isotopes is very complicated, but comes down to energy. In theory, you can just keep piling up neutrons one after anther on an atom's nucleus, and keep making new isotopes. In practice, however, as you add more neutrons you make the nucleus more and more unstable (another way of saying this is having more and more internal energy), because it just has too many neutrons for the number of protons. They don't have to be perfectly balanced (for example, in the previous paragraph I mention how uranium has more neutrons than protons), but you also don't want to have it too unbalanced. Atoms that are unstable this way are called "radioactive", because they try to become stable by giving off energy (radiation) and in the process turning some of those neutrons into protons, or vice-versa (to try to balance things out). For example, when you add one neutron to hydrogen, you get deuterium, which is stable (but rare), and if you add one more neutron you get tritium, which is unstable (radioactive). If you were to add a THIRD neutron, you'd get something that is so extremely unstable that it doesn't even last a billionth of a second - you just can't stick the neutron on, in other words.
Sometimes you can get something that is very unstable, but still lasts for a short period of time, and in total there are over a thousand isotopes, and only a few hundred of them are stable -- the rest lasting for various periods of time before they become stable themselves.