Historically derived from turkish borek, it has grown into an institution.
It's just filo dough, but made fresh overnight, in many layers, filled with urda cheese (made from whey), richly sprinkled with oil, and baked in an oven, in a pizza-sized pan. The burek joints (buregdžinica - the -dži suffix being also turkish, and k being transformed into g before an aspired consonant) have special two-handed knives to cut these into quarters. Burek and jogurt (yoghurt) is the traditional breakfast at work, but also after an all night party. The bakers usually open very early in the morning, so frequently a party extends for an extra hour or two, because someone knows a terrific burek two corners away, opens at 5:00.
There's also burek with grated apples, with ground meat, and with spinach. The apples and meat are regulars, but are made in smaller quantities and they may not have them at all times; the spinach is a rarity. These days, various joints are inventing new kinds of burek, crosbreeding it with pizza frequently.
One should not, under any circumstances, get into a dispute with a Bosnian about what is burek and what is not. They have their own set of names and definitions and enforce them vigorously.