Literally, a fry-in. An addition to almost any stew, or other kinds of boiled dish.
It takes a bit of oil or grease, diced vegetables (parsley leaves, onions, carrot), salt, and it's briefly fried in a pan or a small pot until the veggies have softened (or onions gone glassy and limp), then a spoonful of flour is added "to make it thicker" (the common excuse), and then it's all poured into the big pot. This is when most of the spices go in.
As this is usually the last step when making a čorba, "zapržiti (nekom) čorbu" (to fry-in (someone's) čorba) means to do someone in at the end, or to make one's life hard.
Onion is pretty much the most important ingredient here. Serbian cuisine may be too much of a mix of the surrounding former conquerors' cuisines, but the overwhelming smell of the fried onions in a zaprška for a goulash or a paprikaš is its hallmark. Even literally - if you walk down the staircase in a novogradnja around noon, you should sense this. Just like the smell of garlic is a way to find a restaurant in the States.