It's the same old dinar as it ever was for the last couple of centuries, or since 1214, if you take the medieval (i.e. pre-turkish) times. Well, not quite the same one, it was reissued several times. The whole history is here.
The current dinar is basically the leftover from Sloba's runaway inflation, when it replaced the old SFRY dinar (which was denominated several times, losing first one zero, then four, then six) at the exchange rate of 13 million to one, which lasted two days, then was replaced with 12 million to one. The old dinar vanished in a few months, and the new one was pegged to the german mark. However, this lasted only a few months; in the next five years it went from 1:1 to 14:1. Assuming that the switch to euro was at 2 DEM : 1 EUR, it now stands at about 235:1. However, most of the loss of value happened in Sloba's years; after 2000 the exchange rate changed far slower.
While RSD is rather stable last dozen years, it's still in the habit of people to speak in euros at times - depending on the item - without naming the currency, it's implied. It can be quite confusing, but with the exchange rate being about 118 RSD :1 EUR, you only need to know the order of magnitude to guess which currency it is. Drywall - the majstor will charge you seven (must be euros, nobody will work for 7 RSD per square meter). You may hear "paid two thousand, awful" - ah, it's 2.000 dinars for a bag of groceries, too much, and "paid two thousand, real bargain" - then it's 2.000 EUR for a used car.