Summary: BC is Before Christ. AD is Anno Domini (Year of the Lord). It's years from Jesus's birth (but there's no Year 0). In the Middle Ages Christian countries used different times to start the year, sometimes Christmas, sometimes Easter, sometimes the Roman calendar (made by Julius Caesar) of 1st Jan. They ended up all using 1st Jan for ease, in the 16th-19th centuries.

Long Version:

In English, we don't use "AC" we use "AD". (But for all I know other languages use AC? I can't comment on those, so the following is about English. Italian uses AC, "avanti Christo" which means Before Christ. So the Italian AC is the English BC, not AD.)

AD stands for "Anno Domini"; a latin phrase meaning "The Year of the Lord". It means years since Jesus was born (though there's no year Zero; his first year of life is 1AD. We today tend to say a baby is 1 after a year, not during that year. But see below regarding his birth.)

Originally, the Roman Empire generally used the current Emperor's reign for dating; the 3rd year of Emperor Tiberius, etc. Other kingdoms nearby (such as European ones) used similar ad-hoc systems with their own kings.

However, the Church needed to use some kind of unified calendar to organise Easter celebrations at the same time across multiple lands. They'd been using one devised by the Emperor Diocletian (a famous oppressor of Christians).

In the 6th century, a Monk living in Rome, named Dionysius Exiguus, devised a new calendar that would start with the year of Christ's birth. So the year he believed Christ was born* was 1AD (as a monk working in Rome, he wrote of course in Latin).

*(The year is probably wrong. Luke and Matthew give contradictory historical information regarding the year, and of course they don't use the BC/AD system as it hadn't been invented yet, so it's difficult to tell exactly when Jesus was born. Matthew says he was born before the death of Herod the Great, which was 4BC, while Luke puts Jesus's birth during the census of Quirinus, which was 6AD. It's not surprising, as they wrote 70-90 years after his birth, and their focus in writing about his birth was to say where he was born, not when. That's not important to them.

Dionysius never explained why he believed Jesus was born in the year he called 1AD, so we can't check his calculations.)

This system wasn't originally used widely, but from the 8th century it became more popular. The Venerable Bede used it in his history of the Anglo-Saxons.

So that's the year, but what about the date?

Julius Caesar had made the calendar start on January 1st, because that's when Roman Consuls started their (1 year) term. This continued through most of late antiquity in Roman-based societies. However, by the Middle Ages, this was considered by a lot of people to be somewhat pagan. They indeed thought it was more sensible to use Easter or Christmas. Easter of course, moves around, and you need a specific date for years to start for administration purposes (sometimes March 25th was used). Meanwhile, in the Byzantine Empire and other Orthodox countries, September 1st was often used.

All in all, it was a bit of a mess. For example, Anglo-Saxon England started the year on the 25th of December. When the Normans invaded, they changed it to the 1st of January. Then in 1155, it was moved back to the 25th of December, and this was used until 1752.

This could cause confusion when interacting between different countries. In the 1400s, Russia was 6 months behind Greece! So for 6 months of the year, it was a different year.

As trade and administration grew, this was more and more awkward.

In 1582, the Catholic Church reformed their calendar, due to mistakes in the way the Julian Calendar calculated the date (due to the difficulties in working out leap years) that had thrown Easter out. The Church wanted to make sure they got Easter at the right time. So the Pope moved the date 10 days forward, and changed when leap days were inserted (for example, there was no leap year in 1800 despite it being divisible by four, because the year isn't exactly 365.25 days long).

This year began on January 1st, just like Julius Caesar's calendar that it was a revision of.

Most Catholic societies quickly adopted it, but others didn't. However, as the annoyance of having different calendars added up, which became worse as countries became more interlocked due to trade, and was also very important for coordination and sea travel,, most Protestant countries adopted the Gregorian calendar in the 17th century and 18th century. Prussia adopted it in 1610; the Swiss, Danes, Norwegians and Dutch in 1700; Britain in 1752.

In the 19th and 20th centuries, with the world dominance of European powers, other nations across the world adopted this system. Japan adopted it in 1873, Egypt in 1875, China in 1912, Russia in 1918, Turkey in 1926.

Today however, many countries have 2 calendars; the Gregorian calendar for business and interaction with the world, and their own calendar for religious festivals.