(mostly written in 2013 except update paragraphs)
Even though the instant coffee (nicknamed "nesulja" or similarly, after nescafe) is very frequent now, and there are many cafés where your choice is that (hot or cold) or an espresso, and many people and offices now make filter coffee ("filteruša"), the classic norm calls for turska kafa (Turkish coffee).
update 2019: turkish coffee is no more, it's now called domaća (domestic) everywhere. True, the Turks brought the coffee here, but they mostly drink tea. There's now also a 3-u-1 (three-in-one), which is some mix of nes, espresso and something else, didn't bother to try.
Which will take some instruction. Turska is made by pouring a tall teaspoon (actually called kafena kašičica - little coffee spoon) of finely ground coffee (grade 10 to 12 on your grocery's grinder) over a džezva of boiling water just taken off the hotplate. Not filtered at all; the fine grains fall to the bottom, dissolving while passing through 99o
C water. Some foam remains floating.
You sip it while it's still hot, or wait for it to cool down, depending on your preference. The cup isn't large
(and in Bosnian case, it's a fildžan, without a handle but in a copper holder), so you can take your time.
Sugar is optional - most people go for one or two cubes (which were traditionally served on the platelet by the cup). It's common for the cook to ask what do you drink
- sweeter, medium or bitterer (extremes are always in the comparative, never the positive). If they do ask, then they'll probably make coffee in two pots, adding sugar into cold water.
Some, like your guide, take the "optional" literally; some pour three or four spoons
The important thing is to detect when you've touched the bottom. Then just stop, don't try to drink that, that's talog (the sediment), aka sac (suts). Many people just turn their cup upside down, and what liquid remained in the cup will push the talog down the walls, creating patterns. When this settles, you ask someone to look into your cup - which is a form of tarot or crystal ball gazing. Depending on their imagination, they may guess they saw something in the talog, usually a trip, a person of certain features etc. It's a pastime that nobody takes (too) seriously.
When buying coffee, there are degrees. The most traditional way was to buy green beans, roast them yourself, then grind them by hand immediately before use, to preserve the maximum taste, and have a reason to sit and gossip with your neighbor while you grind. Next one was to buy roast beans and grind it at home - not by hand as much, it became regular to have electric
grinders at home (which were speciall, grade 10 or 12, extra fine - your filter coffee is grade 5 or 6). You could have them ground in the grocery - they all had grinders - and then reseal the bag and take it home. Sometimes you'd get a bag of roast beans as a gift and take it to your grocery to be ground.
Nowadays I see mostly ground coffee on the shelves, and no grinders to buy - I assume that domestic production is shut down, and nobody abroad makes these fine grinders anymore. Also haven't heard the sound of the grinder in the supermarket
in quite a while. Here's the classic coffee grinder, which you can not find today: