What you're referring to is the Anno Domini calendar era. I know it's not really the done thing to quote Wikipedia, so I'll just say that it has a very good summary of its creation by Dionysius Exiguus, bearing in mind that he didn't leave his workings and there are various theories about how he came up with this particular date - the only thing everyone seems to agree on is that he got the date of Jesus's birth wrong. The first year of Anno Domini was 532, but Europe was far from centrally organised at that time, and it wasn't until the rise of Charlemagne that it started to be widely used, largely thanks to Alcuin of York. His time at the Carolingian court was from 782 to 790, so we can set that as a start date for adoption, but it took a long time for it to be universally adopted in Europe. Portugal didn't switch to AD until 1422 and some Eastern European countries were still using the Byzantine calendar into the 20th century. Further afield there are still other calendars in use, of course. In Beijing both the Chinese and the Gregorian calendars have existed side by side for the last couple of centuries, and which one you use depends on cultural context.

If the Gregorian calendar (currently international standard - Pope Gregory XIII) came about in 1582, what was used before that?

Add on question: Are there countries that don't use the Gregorian calendar or the AD/CE numbering system?