There is no year zero. It goes 2 BC, 1 BC, 1 AD, 2 AD, etc.

The dating system we use was developed by a monk named Dionysius Exiguus. From about the beginning of the 6th century he lived in Rome (he was probably from Scythia Minor, ie Romania/Bulgaria, originally). He did a lot of Greek to Latin translations.

Dionysius wanted to calculate when, precisely, Easter should be, as the Romans and Greeks didn't always have the same date. He also wished to replace the current (Roman) year system used with a Christian system. So, in the year of "the consulship of Probus Junior" he announced that it was 525 Anno Domini (along with various calculations about Easter, leap dates, etc). His system was quickly adopted by the Church for calculating Easter. However, it was still not actually used by most people to give the year, and they continued using whatever year system they'd used before (if any).

It's only really in the 8th century that it began to be widely used. For example, the Venerable Bede, who wrote a history of the English in the 8th century, used it.

You may also wonder, as many have argued that 1AD is not when Jesus was born, how Dionysius worked out the year was 525 years after Jesus' birth. And the answer is... we don't know. He explained how he calculated things like Easter, but not why he thought Jesus was born in 1AD.

So, to answer your secondary question, no, people in 500AD didn't know it was 500AD. Some people in 600AD would have known it was 600AD, but it wasn't widely used. Whereas most people in Western Europe in, say, 1000AD, would have known that it was 1000AD.