One thing to clarify about the "twins run on the mother's, not the father's side" thing: by that, they mean the mother of the possible twins (as opposed to the father of the possible twins). So, for the potentially-pregnant woman in question, both *her* mother and father's side are contributors to chances of fraternal twins. Basically, the main thing that runs in families is a propensity to have multiple eggs ovulate in a single cycle/month.
With identical twins, the conventional wisdom is that these are one-off flukes and that the trait is not passed down. However, some people do think there may be a slight genetic component (in either the mother's eggs or the father's sperm, or both) that adds to the possibility of the embryos splitting (and hence having identical twins). The evidence for this is still out, though, (beyond the anecdotal level) and if it is the case, it is less of a possibility than with passing along fraternal-- but who knows!
(And of course this whole conversation assumes no fertility treatments, which are a whole different story! Though I am constantly amused by the number of people-- who know for a fact that I had many, many rounds of fertility treatments of various sorts-- who ask me if twins run in my family, when they find out I'm expecting two)