AD stands for Anno Domini meaning "In the year of our Lord".
AD (or A.D.) is an abbreviation for the Latin expression "Anno Domini", which translates to "the Year of Our Lord", and equivalent to C.E. (the Common Era). Anno Domini refers to the years which followed the supposed birth year of the philosopher and founder of Christianity, Jesus Christ. For the purposes of proper grammar, the format is properly with the A.D. before the number of the year, so A.D. 2018 means "The Year of Our Lord 2018", although it is sometimes placed before the year as well, paralleling the use of B.C.
The choice of starting a calendar with the birth year of Christ was first suggested by a few Christian bishops including Clemens of Alexandria in C.E. 190 and Bishop Eusebius at Antioch, C.E. 314–325. These men labored to discover what year Christ would have been born by using available chronologies, astronomical calculations, and astrological speculation.
Dionysius and Dating Christ
In 525 C.E., the Scythian monk Dionysius Exiguus used the earlier computations, plus additional stories from religious elders, to form a timeline for Christ's life. Dionysius is the one credited with the selection of the "AD 1" birth date that we use today—although it turns out he was off by some four years. That wasn't really his purpose, but Dionysius called the years that occurred after Christ's supposed birth "The years of our Lord Jesus Christ" or "Anno Domini".
Dionysius's real purpose was trying to pin down the day of the year on which it would be proper for Christians to celebrate Easter. (see the article by Teres for a detailed description of Dionysius efforts). Nearly a thousand years later, the struggle to figure out when to celebrate Easter led to the reformation of the original Roman calendar called the Julian Calendar into the one most of the west uses today--the Gregorian calendar.
The Gregorian Reform
The Gregorian reform was established in October of 1582 when Pope Gregory XIII published his papal bull "Inter Gravissimas". That bull noted that the existing Julian calendar in place since 46 B.C.E. had drifted 12 days off-course. The reason the Julian calendar had drifted so far is detailed in the article on B.C.: but briefly, calculating the exact number of days in a solar year was nearly impossible prior to modern technology, and Julius Caesar's astrologists got it wrong by about 11 minutes a year. Eleven minutes isn't too bad for 46 B.C.E., but it was a twelve-day lag after 1,600 years.
However, in reality, the main reasons for the Gregorian change to the Julian calendar were political and religious ones. Arguably, the highest holy day in the Christian calendar is Easter, the date of the "ascension", when the Christ was said to have been resurrected from the dead. The Christian church felt that it had to have a separate celebration day for Easter than the one originally used by the founding church fathers, at the start of the Jewish Passover.