Durum Flour is an unbleached flour ground from the hard wheat called "Durum wheat."
Though Durum wheat is used to also make semolina, semolina isn't the same as Durum Flour.
Semolina is made by coarsely grinding the endosperm of Durum wheat. While grinding it, fine powder is also produced. This fine powder is what is called Durum Flour. So, in theory, it's a by-product of making Semolina, though Durum Flour is of course also purpose-made by double-milling the wheat into a fine flour.
Durum Flour is used for pasta, noodles, couscous and some speciality breads. Home pasta makers often prefer working with Durum Flour as opposed to semolina, as it is easier to work with -- Durum Flour mixes easier with water, and forms a smoother dough.
Durum Flour is often preferred, though, for noodles as opposed to pasta, because it is more finely-textured. It is also hard and glutinous.
Durum and Semolina Flours have the highest protein content of any wheat flour. Despite that, the gluten that forms in them is weak. This makes them good for pasta and noodles, as the dough doesn't get tough as you work it -- it stays tender.
Bread made with Durum Flours tends to be yellowish-coloured, have a taste distinct from breads made with other flours, and have a longer shelf life.
Durum gets its name from the Latin word for "hard".