Elder Uchtdorf told us to read several sections in the Doctrine and Covenants on the priesthood so we'll be taking a brief hiatus from our normal exploration of the Book of Mormon while we do as he asked.
D&C 20:1 The rise of the church of Christ in these last days, being one thousand eight hundred and thirty years since the comin gof our lord an dsavior Jesus Christ in the flesh. James E. Talmage says in Jesus the Christ that "We believe that Jesus Christ was born in Bethlehem of Judea, April 6, BC 1" and he explains his reasoning by citing the fact that there is not general consensus among biblical scholars about a date, and quoting this verse in D&C 20, "In which the day is made without qualification the completion of the one thousand eight hundred and thirtieth year." There is no concensus among biblical scholars (concensus is lacking among both LDS and non-LDS scholars). Talmage references Dr. Charles F. Deems' book, The Light of Nations and quotes, "the birth of our Lord is placed in BC 1 by Pearson and Hug; BC 2 by Scalinger; BC 3 by Baronius and Paulus; BC 4 by Bengel, Wieseler, and Greswell; BC 5 by Usher and Petavius; BC 6 by Strong, Luvin, and Clark; BC 7 by Ideler and Sanclemente."
This one is interesting because according to Josephus, a non-christian Jewish historian who wrote at this time, King Herod (the one who killed all the babies because he was afraid of Jesus's impact on his dynasty) died in BC 4, making it difficult for him to have killed anybody after that. (Josephus records an eclipse of the moon shortly before Herod's death, which astronomers date to March 12 or 13 of 4 BC) Our current calendar is based on the research of a fellow named Dionysius Exiguus who worked it out in 532 AD based on the date of the founding of Rome.
There's one other monkey wrench to throw in there: the Census of Quirinius was taken in 6 AD, and is the only one close to this time. If that census was taken in 6, and Herod died in 4, I like 6 but I'm certainly nothing like a scholar, and I'm sure there are whole books dedicated to this that talk about way more than I'm finding.
In counterpoint to James E. Talmage (as they usually seem to find themselves), Bruce R. McConkie wrote "We do not believe it is possible with the present state of our knowledge-including that which is known both in and out of the Church-to state with finality when the natal day fo the Lord Jesus actually occurred." (McConkie, 1:394, n.2)
The significance of the implication that the ,day is April 6 is meaningful for LDS church history. On April 6, the prophet Joseph Smith restored the church (1830), The cornerstone of the Salt Lake Temple was dedicated (1853), the St. George temple was dedicated (1877), The Salt Lake Temple wa dedicated (1893). But, I'd bet you could find three or four meaningful events on pretty much any date you picked if you're looking that granularly.