The calendar used in our society was invented by a monk named Dionysius Exiguus, and was originally intended to reliably reckon what day Easter should be celebrated each year.
There is a great explanation of how the meaning of the “AD” and “BC” labels relate to counting in the book “Marking Time: The Epic Quest to Invent the Perfect Calendar” by Duncan Steel
Mathematicians distinguish between cardinal and ordinal numbers. Cardinal numbers are simply labels for quantities, like one, two, three. Ordinal numbers refer to position in a sequence, such as first, second, third. Our year-counting scheme employs ordinal numbers. Thus A.D. 1 implies the first year of the life of Jesus Christ (as reckoned from the traditional date of his naming and circumcision, on January 1 in that year). The previous year is termed 1 B.C. because it was the first year before A.D. 1(i.e., an ordering is implied). In such a system, clearly the concept of a year zero has no meaning: one cannot have a year which was zero years before or after an event. The astronomers' system of numbering years (-3, -2, -1, 0, 1, 2, 3) only makes sense because cardinal numbers are used.
To paraphrase: A.D. 1 refers to the one year period that immediately followed Christ's naming and circumcision, and 1 B.C. refers to the one year period that immediately preceded Christ's naming and circumcision. These are "the first year of Christ" and "the first year before Christ". There was no year in between.
Anno Domini is translated as “in the year of the Lord”. The first anniversary of Christ’s naming and circumcision occurred AFTER the passage of his first year of life. That first year of his life (before the first anniversary) is what “A.D. 1” refers to in Dionysius Exiguus’ system. So the END of the first year of Christ’s life marks the end of A.D. 1, and the beginning of A.D. 2. That is, the beginning of A.D. 1 marks the beginning of Christ’s life, not the end of his first year of life.
The religious “correctness” of the calendar is a separate issue, as the numbering of the years represented Dionysius Exiguus’ ATTEMPT to estimate the year of Christ’s birth relative to the calendars already in use at the time. It is widely believed that Dionysius Exiguus was incorrect and that Christ’s actual birth year was earlier than the year Dionysius Exiguus labeled “A.D. 1”.
If anyone is still awake, there’s also some interesting info on the topic here:
http://www.tondering.dk/claus/cal/node3.html
In about AD 523, the papal chancellor, Bonifatius, asked a monk by the name of Dionysius Exiguus to devise a way to implement the rules from the Council of Nicaea (the so-called ``Alexandrine Rules'') for general use.
Dionysius Exiguus (in English known as Denis the Little) was a monk from Scythia, he was a canon in the Roman Curia, and his assignment was to prepare calculations of the dates of Easter. At that time it was customary to count years since the reign of emperor Diocletian; but in his calculations Dionysius chose to number the years since the birth of Christ, rather than honour the persecutor Diocletian.
Dionysius (wrongly) fixed Jesus’ birth with respect to Diocletian’s reign in such a manner that it falls on 25 December 753 AUC (ab urbe condita, i.e. since the founding of Rome), thus making the current era start with AD 1 on 1 January 754 AUC.
How Dionysius established the year of Christ’s birth is not known (see section 2.14.1 for a couple of theories). Jesus was born under the reign of King Herod the Great, who died in 750 AUC, which means that Jesus could have been born no later than that year. Dionysius’ calculations were disputed at a very early stage.
When people started dating years before 754 AUC using the term ``Before Christ’', they let the year 1 BC immediately precede AD 1 with no intervening year zero.
Note, however, that astronomers frequently use another way of numbering the years BC. Instead of 1 BC they use 0, instead of 2 BC they use -1, instead of 3 BC they use -2, etc.