Comes from "čekati" - to wait. "Dočekati" is a finite version of the verb, most precisely translated as "wait until", "to have waited until", or "wait to see it come/happen".
Doček is therefore an awaital ceremony - you go to the station to dočekati a guest, or you have a party to dočekati the New Year. Pretty much the opposite of a farewell party. The New year doček is, of course, The Doček.
Since Christianity was out of fashion for the 2nd half of the XX century, and the Serbian Christmas is on 7th of january anyway (the local church being probably the last one to stick to Julian calendar), and the winter marathon is already rolling, New year's night, i.e. The Doček, is the most important party of the year. The preparations start about two weeks ahead, people start stocking up the drinks, sour cabbage (for rasol will be needed to cure the hangover), piglets are slaughtered. The last two days the shops are chock full of people - parents are buying gifts for their kids, the new year firs are hauled (never had much of a tradition for Christmas trees anyway, but built a big one over NY). About ten days ahead, people start asking each other "where will you [go] for doček?", and some are in a state of panic if their answer may turn to be "don't know".
The doček itself can be anywhere, and organized by anybody. Traditional types:
- "modestly, in family circle" - the whole family sits around the table, eats, drinks and watches TV. Occasionally, they sing. Sometimes, there are relatives visiting.
- at the office - falling out of fashion, because very few enterprises still have their own kitchens, so food would have to be brought by some catering service. Most of the people would come with their families (kids left to grandparents), there'd be a long table, or two long tables, with space in the middle for dancing (a kolo would be mandatory), there'd be a small band of maybe just one accordeon player, and everyone would get drunk, some severely.
- private parties - the teenagers would arrange for the host's parents to go somewhere on their own, so they'd have the crate ("gajba" - the apartment or house, as long as no adults are around). They'd stock up the drinks, maybe ease up on the young pork and sour cabbage and go for sandwiches and russian salad.
- party in a restaurant or a hotel or just about every watering hole - they'd have a band, which would play everything from hotel-wave pop to old city romance to turbo folk, and suck at everything (by the old jazz principle "it doesn't matter what you play, it's how you play", they'd play everything equally bad), but anyone who got into their Balkan drunken night mode would be stuffing the accordeon with the banknotes. There'd be dancing room between the tables, and the kolo would snake between them regardless of where that space was. The consummation fee would differ, depending on the place's fame, and it generally includes food, two drinks, and music. After two drinks, the tab starts running, which may cost a lot but nevertheless pretty much everyone gets more or less drunk.
- outside party on a square. One would say that's for losers who can't have a regular doček, but then that's also for those who had enough of the same old every year, and for those who go party hopping, and somehow arrange to appear there around midnight, then go on partying. These are often organized by cities themselves - tax money pays for a band to play outside (probably with heaters on the stage, it's usually quite freezing then). The atmosphere is emphasized by lots of firecrackers firing off everywhere, despite the crackdown the cities do on the sellers weeks ahead.
What's common to all of these: at midnight, everyone kisses everyone else. Champagne (or other sparkling wine - local brands "Milion" and "Fruškogorski biser" were popular for decades) is popped open and drank quick. Windows are opened and firearms discharged in the air, so at moments it sounds like Baghdad or Beirut at its worst.
The parties end anywhere between 1am and 7am, depending on the stamina of participants.