There are two kinds - the intercity (međugradski) and local, aka city buses. They are separate systems, have separate stations and frequently any transportation company does only one of thes. Actually, there's the intermediate step, the interplace (i.e. short distance intercity) which usually connects a city with villages around it. That may be covered by any company.
In the times of SFRY, the transporters were self-managed enterprises just like any others, and they were competing on the market. In intercity buses, this led to various bad practices - speeding up to appear at the next stop before the competitor's bus does so to pick its passengers; using any convenient place as a stop (frequently a tavern), inventing weird bus lines just to spite the competitors (if the buses were full at 15:00 from A to B and the slot was taken so no new buses on that line allowed, they'd invent a line going at 14:45 from a village before A to another village after B, so it's formally not the same line), and specially the habit of selling more tickets than there are seats.
The bus wars started dwindling down by mid-seventies, when the whole field got far better regulated and controlled by traffic militia. If you got into a bus on one of the few remaining curb stations (if the regular station is at the east end, then there'd be one spot allowed near west, north and east exits) and there were no seats, you still got to pay the ticket and have the seat when one is vacated after the bus passes a couple of villages. However, when that didn't happen, the drivers knew where the patrols were so anyone standing in the aisle was obliged to duck.
Nowadays, most of those regulations are still in place, but no more bus wars. The market is drying up, and on lines where you had 10 rides a day, you may find only four. The short distance buses are now smaller than regular buses, because the cars are cheap and there aren't that many village kids to go to city high schools and there are far fewer jobs so commuters with employer-reimbursed tickets are also few.
There's no [such thing as] Greyhound. Perhaps Lasta (swallow), the largest and oldest company, but their market share can't be more than 1%. There are too many small local companies, every city has at least one, plus there are tourist companies with buses which don't do line traffic. They haul their customers.
There's another phenomenon - the cab drivers. They usually have their own station at every bus station, and they freely offer a ride at the price of the bus ticket (usually without the station ticket, that each station charges separately), provided that you wait until they have three or four passengers. They won't drive you to the station there, just close enough to catch some local transportation - but they'd be faster (unless you wait too long) than the bus, and they don't have the nearly mandatory turbo folk playing. Nowadays, though, the odds of having to listen to it on the bus are about 50:50. Back in the day it was almost 100:0.