It's pretty complicated, but there are two factors that determine whether an isotope is stable or not:

1) neutron-to-proton ratio, and

2) atomic number

Let's address the second factor first. Generally, there are no stable isotopes with an atomic number greater than 82 (which means that lead is the heaviest element that has any stable isotopes). It was once thought that bismuth-209 was stable, but now scientists believe that it is radioactive with an incredibly long half-life. (Still, some scientists refer to Bi-209 as being "virtually stable").

If an isotope's atomic number is 82 or below, its stability is dictated by its neutron-to-proton ratio. If the neutron-to-proton ratio is too low or too high, the isotope will be radioactive. How do you determine the ideal neutron-to-proton ratio for an element? Unfortunately, it's not so simple. Lighter stable isotopes tend to have neutron-to-proton ratios close to 1. (For example, the neutron-to-proton ratios in the stable isotopes C-12, N-14, and O-16 are all exactly equal to 1). As the atomic number increases, so does the most stable neutron-to-proton ratio. The neutron-to-proton ratio for lead-208, the heaviest stable isotope, is 1.53 to 1.

Furthermore, there is a bit of wiggle room in the neutron-to-proton ratios of the stable isotopes. Although most elements with odd atomic numbers have only one or two stable isotopes, even-numbered elements generally have multiple stable isotopes. Tin, element 50, has the greatest number of stable isotopes of any element, with ten.

There are magic numbers to help you determine whether an isotope is likely to be stable or not. Generally speaking, isotopes that have 2, 8, 20, 28, 50, 82, or 114 protons should be particularly stable (relative to surrounding isotopes, that is), as well as isotopes that have 2, 8, 20, 28, 50, 82, 126, or 184 neutrons. (Note that some of these atoms are so huge that they exist only in theory.) Also, an isotope that has even numbers of protons and neutrons is much more likely to be stable than an isotope that has odd numbers. There are 163 stable isotopes that have even numbers of protons and neutrons, 53 that have an even number of protons and an odd number of neutrons, 50 that have an odd number of protons and an even number of neutrons, and only 4 that have an odd number of protons and neutrons.

EDIT: Simply because a nucleus is magic (meaning it has a magic number of protons or of neutrons) does not mean that it will be stable, but it will be *relatively* stable compared to similar non-magic nuclei. For example, carbon-14 has 8 neutrons - a magic number. C-14 is radioactive because it's neutron-to-proton ratio is too high, but its half-life is more than 5700 years. This is far, far longer than the half-lives of C-15 (2.45 seconds) or C-16 (0.65 seconds). Likewise, element 114 (flerovium) is also radioactive; however, its longest-lived isotope, Fl-289, has a half-life of 21 seconds. Compare that to the half-lives of the longest-lived isotopes of element 113 (0.4 seconds) and element 115 (0.05 seconds).

At the end of the day, however, the quickest, easiest way to determine whether an isotope is radioactive or not...is to look it up.

I hope that helps. Good luck!