The short answer is that durum flour is flour which is ground from Durum wheat (also referred to as Semolina). Durum wheat is a type of wheat that has an especially high protein content, and in fact its name derives from the Latin word for "hard".
Durum flour, with its high protein content, makes dough with unusually strong gluten. It's generally used in pizza dough and pasta, and its more coarsely ground cousin semolina is used to make cous cous. Though you can make bread with durum flour, recipes usually mix durum and regular wheat flour.
Durum wheat can be ground into semolina (sometimes called semolina flour if it's ground more finely) and durum flour, which can be referred to as extra fancy durum flour or extra fancy pasta flour. However, you can't really substitute one for the other. Semolina is a coarse grind and it looks a lot like corn meal. In fact a lot of bakers and pizza makers use semolina to dust baking sheets and peels so the dough doesn't stick (check the bottom of your pizza next time). Durum flour is ground as fine as regular flour and that's why it can be used in bread and pasta doughs.