Farina is made from hard wheat that is not durum wheat. It is similar to semolina, but semolina is made from durum wheat.
Just the just the endosperm of the wheat kernel is used. The bran and most of the germ are removed. The endosperm is then ground finely and sifted. It'snot ground as finely as today's white flour is, though: it's more granula, like a meal. It looks like a paler version of cornmeal. It is in fact about the same coarseness as cornmeal, making it coarser than semolina.
Farina is a cheaper flour than semolina. In Italy, it's illegal to use farina commercially to stretch out the semolina when making pasta. It's not in North America. Pasta with farina in it isn't considered as good a quality.
Semolina is possibly what Hungarian recipes mean when they call for "grizes liszt" (gritty flour, "liszt" meaning "flour" in Hungarian.) Cream of Wheat or Farina can be used as a substitute for "griz" in Hungarian recipes (see Language Notes below.)
Farina can be used as a thickener in dishes where the texture won't matter so much, such on the top of fruit pile fillings. It can absorb a good deal of moisture and juice, to prevent top crusts from getting soggy.
You can cook Farina up as a porridge for breakfast and serve it in a bowl, or cook it a bit thicker and treat it like polenta for a meal later in the day. When cooking it a porridge, Farina cooks up less fluffy than semolina.
You can also use Farina as a savoury side grain, as you would rice: In a frying pan or sauce pan, brown Farina in a bit of oil. Add cold water, salt and pepper to taste, and let simmer over medium heat until water has been taken up and the grain can be fluffed. Per 1 part Farina, use 1 part water (e.g. for 1 cup of dry Farina, use 1 cup of water).
Literature & Lore
"Farina is one of the most delicate of cereals for invalids and children; it is more easily digested than the whole grains rolled. It does not, however, contain as much nourishment.
Put one quart of boiling water in a saucepan directly over the fire; add a level teaspoonful of salt, and then sprinkle in slowly a half pint of farina, stirring all the while. Boil rapidly for ten minutes, then stand in a pan of boiling water, or in a double boiler, and cook twenty minutes longer.
Left-over farina may be cut into blocks, dipped in egg and bread crumbs, and fried; or may be used as a thickening for soups, or in timbales with fruit and cream."
Eastern Europeans call it "griz" or "griess", though the word "griz" is also applied both to what we would think of as groats and to what we call "semolina"