Bosnia and Herzegovina has a very strong cuisine for a country of its size. In general, I would say it's very 'homey', with a lot of slow cooked stews and meats with delicate spicing - it is said that Bosnian cuisine centres on preserving the natural flavour of each ingredient to the extent possible. Much of what you find is not unique to the country but part of a regional cuisine, with many Ottoman influences playing out today. (Though, no doubt Bosnian cuisine has its own nuances.)
One issue, I would say, and because people don't seem to like their food being messed with, is there is not a great deal of variation. Hence the existence, for example, of just three or four staple burek varieties. In many other places you’d have a chicken tikka masala one by now, or a morocan mint and lamb one. Not in Bosnia. Similarly, you won’t get tzatziki with your cevapi, even though it would pair wonderfully - just the onions and maybe some yoghurt.
There are a range of sweets that are made in Bosnia too. Again, there is a lot of influence here from the Ottoman's and delicacies such as Turkish delight and baklava are common. In general, there is a heavy focus on items drenched in honey or syrup. It's worth noting here that Bosnia and Herzegovina is said to produce very high quality honey.
Finally, Bosnia and Herzegovina has a long tradition of producing wines, spirits and beers. The list of types of rakija, for example, is genuinely astonishing and many families produce their own on an annual basis and bottle it up for home consumption and for friends.
If you want to learn much more I recommend a book called Bosnian Cuisine (ISBN: 9789958180743) which you can find in some stores in Sarajevo.
Savoury foods
Ćevapi
Ćevapi is apparently considered to be the national dish of Bosnia and Herzegovina. It is essentially, a mixture of mince meat and spices, formed into oblong sausages and grilled. It is most often served with somun bread and chopped raw onions. Often when you order it you will be asked if you want five pieces or ten pieces. The best, supposedly, is made in Travnik.
Burek or pita
Burek, often called pita, is a stuffed pastry dish. In Bosnia it is most often filled with one of four types of filling. It is most often made by rolling the filling into the pastry in a tube shape before spiralling the tube. The dish is then cooked under a 'sač' (a large heated bell). Others are made more like a lasagne, layering the pastry horizontally. The most common options are: (i) Sirnica - cheese; (ii) Burek - meat; (iii) Krompirusa - potato; and, (iv) Zeljanica. Typically you buy it by weight. Burek/pita can be eaten for any meal of the day: I most often had it for breakfast with yoghurt.
Meat
Meat is a favourite in Bosnia and Herzegovina. Often a big plate of meat will be served with potatoes, a few sides, and not much else. So you don't get lost, the Bosnian names for the different meats are: jagnjetina = lamb, teletina = veal, govedina = beef, piletina = chicken, svinjetiva = pork, kobasica = sausage, suho meso = dried/cured meat.
Ispod sača
"Ispod sača" is a method of cooking and one that is quite common in Bosnia and Herzegovina. "Ispod" means "under" and "sača" translates roughly to "bell". I suppose it's hard to find an exact translation but "lid" might be closer. It is a big metal (sometimes ceramic) lid (which is shaped a bit like a shallow bell) which is heated by placing hot coals on top of it and placed over the food that is being cooked. Here is a video of a meat burek being cooked 'ispod sača'.
Spit roast lamb
Exactly that: spit roast lamb (whole). See the image at the top of this page. This is a very popular dish in Bosnia and Herzegovina and if you drive around the country you will often see them being spit roasted outside restaurants close to the road. Stop for lunch and get some. Here is a step by step video of how to cook "roast lamb the Bosnian way".
Bosanski lonac
This would translate to "Bosnian stew". There's is a wide variation in how these are prepared, but in general it is a rich and slow cooked stew, often with meat. This is another dish that has been described as the national dish of Bosnia and Herzegovina.
Pura
Pura is a popular breakfast dish in Bosnia and Herzegovina. Pura means "polenta" or "oatmeal porridge" and it is served with young white cheese or dried meat. It's a simple but filling breakfast.
More savoury foods
Uštipci
Uštipci could be described as fried dough balls. In Bosnia and Herzegovina they are generally savoury and often served with a thick dairy product called kaymak (which is like a very rich cream cheese) as well as with sliced cheese and dried meat. You cut them open and spread the kaymak on them. You can also stuff them with the cheese and dried meat to make fried dough sandwiches. Here is an uštipci recipe.
Čorba
Čorba means soup or stew or broth. There are many types available in Bosnia and Herzegovina. The most common is perhaps Begova čorba (also called Bey's soup), which is made with chicken, okra, carrot, parsley, egg yolk, and rice.
Klepe
Klepe are like ravioli. They are little minced meat dumplings served with a a sauce of sour cream, butter, and lots of garlic. Take a look at a klepe recipe here.
Dolma
Dolma is a stuffed dish, made by stuffing vegetables (eggplant, zucchini, peppers, onion) with a filling, often mince meat, rice, spices, or other vegetables. They are usually cooked in a pot with a lid at a low heat. They each have a different name. My favourite, for example, is sogan dolma, which is stuffed onion.
Sarma
Sarma is similar to dolma with one key difference: sarmas are made by wrapping vegetables around a filling (mostly cabbage or vine leaves). I would say that cabbage leaves are the most common in Bosnia and Herzegovina.
Šopska salata
Šopska salata is a popular salad across the balkans (said to orginate in Bulgaria). It is made from tomatoes, onions, peppers, cucumbers, and a young white cheese on top. It typically comes with a vinaigrette.
Kupus salata
Kupus salata is cabbage salad. Along with šopska salata, this is the most likely form of salad that you will be served. It is raw grated/shredded cabbage which may or may not have been picked.
Ajvar
Ajvar is a condiment that is made by roasting red bell peppers and eggplants. This is a product that many Bosnians will take a day to make at the end of the pepper growing season at home in their garden with their family and friends, roasting red bell papers on an open fire. Some varieties are made to be spicy and others have a sweeter taste. Take a look at an ajvar recipe here.
Sweet foods
Baklava
Baklava is made by layering paper thin filo pastry (jufkas) and sprinking each layer with finely chopped or ground nuts and baking. A sugar syrup (or honey) is then poured over the baklava. See a Bosnian Baklava recipe here.
Tufahija
Tufahija is really quite special and not something to be tackled lightly. Tufahija is a walnut stuffed apple that is poached in sugar water and often topped with cream. It's very, very sweet. Learn how to make one here.
Hurmašica
This is a pastry made from flour, butter, eggs, and sugar that is doused in lemon-flavour sugar syrup.
Lokum
Lokum is essentially another word for Turkish delight (a family of confections based on a gel of starch and sugar). This dessert is popular in Bosnia and Herzegovina.
Tulumbe
This is a deep fried batter dessert that is soaked in syrup while it is still hot. it is later served cold and has a galzed appearance. In some ways it is similar to churros. Learn how to make tulumbe here.
Honey
Bosnian honey is said to be very high quality, partly due to its relatively low population density and a large areas with limited industry and limited agriculture with large pesticide application. You'll often see small producers selling honey at the side of the road, labelled as "med", the Bosnian word for honey.
Dairy products
Cheese
Bosnia has a decent variety of cheeses (sir = cheese, in Bosnian and is pronounced "seer") and I encourage you to try them. Some of the most common are: Travnički sir/Vlasički sir, which is a soft, white cheese made from sheeps's milk; Sir iz mijeha, which means cheese from the sack, and is produced in a sheepskin sack; and Ljivanjski sir, which comes from Livno and is a firmer, yellow cheese, generally made from cow's milk; Trappist sir, which is produced by monks near Banja Luke and is also a semi-hard, cow's milk cheese. More broadly speaking, Mladi sir means "young cheese", Kozji sir means "goat cheese", and Suhi sir means "smoked cheese". Read about trappist cheese here.
Kajmak
Kajmak has been described as the clotted cream of the Balkans. It's kind of a spreadable young cheese or cream that has a very rich flavour (the fat content is typically about 60%). I've not seen it anywhere else in the world so definitely try to taste it, with uštipci if you can, while you are in Bosnia. Read this article for way more detail on kajmak.
Fermented milk drinks
There a wide range of fermented/soured milk/youghurt drinks available in Bosnia and they are often drunk with pita/burek or ćevapi. You'll likely see ayran, kefir, and kiselo mlijeko, which essentially means soured or fermented.
Alcoholic beverages
Rakija
It would be fair to say that for many, rakija is the national drink of bosnia. A common phrase / joke that is explained to forgeiners in Bosnia is "Rakija: connecting people". Rakija is the "collective term for fruit spirits that are popular in the Balkans". You'll also find some that are infused with herbs, walnut, and honey. Rakija is typically about 40%, but it could be higher, especially if you are sampling someone's homemade.
Beer also has a long history in Bosnia and Herzegovina and there are several commercial scale beer brands (Sarajevsko pivo, Nektar, Tuzlanski pivo, and preminger, among others). More recently, a craft beer scen has been springing up, the two largest of which are probably Gelender and Brew Pub. Check out Rate Beer's ranking of various Bosnian beers.
Non-Alcoholic beverages
Coffee
Apparently Sarajevo opened its first coffee house in 1591, this is before London, Paris, and Vienna, for example. In Bosnia they use a "džezva" to prepare coffee. It often comes with lokum and a sugar cube. You’ll read that coffee has long been a ritual in Bosnia and Herzegovina. In many ways it still is, but it serves more as a way of getting out of the office or whiling away a morning, afternoon, or evening with friends.
Planinski čaj
Planinski čaj translates to "mountain tea". It depends where you get it from, but this tea is an assortment of herbs collected on the mountains, often said to have medicinal purposes (you actually might get asked if you are unwell if you order a tea in Bosnia as it has an association with healing). Bosnian mountain tea might contain birch, thyme, st.johnswort, mint, raspberry leaves, and much more.
