Dionysius Exiguus came up with the present system in the early 6th century. u/sunagainstgold has the background here, plus there's some good follow up questions and discussion in the comments: https://www.reddit.com/r/AskHistorians/comments/5izv93/comment/dbc9y1r/

Why did a Byzantine monk care about years? Because timekeeping has historically been of utmost importance to the church for calculating the dates of days like Easter. Having a standard reference point for years is obviously much more convenient, and no one's going to argue over using the central figure in Christianity as the reference point. How did he calculate the birth of Jesus (and was it his birth or conception)? No one's quite sure! Based on the calculation he might have been trying to do he may have been off 4 or 5 years - this doesn't mean our calendar is "missing" 4 years (i.e. it's actually 2026), it just means Jesus would actually have been born in 4 BC based on Dionysus' calculation, but again no one knows for sure his method anyways. This gets discussed lower in the comments in that answer.

And one more thing: there is no year 0. There's 1 BC, then there's 1 AD. Technically when everyone was celebrating the millennium in 2000 they were one year early - 2000 was the last year of the 20th century, not the first year of the 21st. Of course, no one cared, getting to start dates with a 2 was neat!