Basically, BC stands for "before Christ" and AD stands for Anno Domini, Latin for "in the year of our Lord." The transition point between BC and AD marks the approximate time when God incarnate came down to earth to live with His creation. (Jesus was actually probably born four to six years earlier.)
Before the development of the BC/AD system of year counting, years were usually identified by their relation to the ruler of that time. We can see this in such passages as Luke 1:5—"In the days of Herod, king of Judea…" and Esther 1:1—"Now it took place in the days of Ahasuerus, the Ahasuerus who reigned from India to Ethiopia over 127 provinces…" This worked fine for local records, but with the influence of the Catholic Church expanding to several different nations with many different rulers, it wasn't practical for church use. The prior dating system also had the unhappy propensity to recognize unpopular rulers in something so mundane as the date.
This was the case when Dionysius Exiguus set out to create a calendar showing the correct dates of Easter. His primary option at the time was to base his years on the emperor Diocletian who had persecuted Christians nearly 250 years before. Instead, he rewrote the calendar, counting up from the incarnation of Christ which he determined had occurred 525 years before (although he left no indication as to how he arrived at that number). Thus, in western Europe, at least, the calendar system was changed from one based on the number of years since a long-dead tyrant came to rule, to the number of years since Jesus was incarnated—"Year of our Lord," which is shortened to "AD." The usage spread erratically; Portugal finally adopted the format in 1422.
"BC" just means Before Christ and counts backward from the same year AD starts (there is no year "0"). BC didn't come into popular use until the seventeenth century. Non-European nations, who based their calendars on their own rulers or on the origin of their countries, have started using BC/AD only in the last century as transportation and trade has become more global. Scientific and secular individuals, not wishing to recognize a religious figure in such mundane things as date, keep the numerical portion of the year for continuity, but change BC to BCE—Before Common Era—and AD to CE—Common Era.
It is interesting to note that by using AD and BC most everyone on the planet marks the date by the (approximate) arrival of Jesus. It doesn't matter if the person is an atheist or comes from a country far from Jesus' birthplace. Jesus, the author of time, is the basis for how we measure time.
Blessed be the name of God forever and ever,
to whom belong wisdom and might.
He changes times and seasons…"
Daniel 2:20-21b
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