The Apocrypha, whose name comes from a Greek work meaning 'hidden', are the set of 14 books of unknown authenticity that have been included in the canonized Old Testament at one point or another through the history of Christianity. The Jewish community has never accepted these books as inspired, and they are not present in the Protestant canon.

St. Augustine was the only significant figure to hold the apocrypha to be divinely inspired, but his voice on the matter was consistently undermined by that of St. Jerome, a considerably more talented linguist and prominent translator. Upon the death of Pope Damascus in 384AD, however, Jerome's influence was recuced drastically. Jerome was the chief secretary to the pope, and aimed to be his successor. When Siricius was elected, however, Jerome moved to Bethlehem, a rather remote section of the Roman Empire, to finish his work on translating the Hebrew Old Testament into Latin Slang. This opened Augustine's opportunity for influence on the councils, and in 393 and 397AD, the councils of Hippo and Carthage finally accepted the apocrypha into the canon. The Syrian Church didn't accept them until the next century.

Most define the apocrypha in terms of the books 'left out' of the Old testament by the Protestants, however, I believe it's important to note that these books had no place in the Jewish canon and were never added to the Christian canon until St. Augustine used the Roman Catholic Church to do so. His efforts in this effect were fought vehemently, but in 450 AD (after 70 years of debating the issue), Augustine finally won out.

Of the reasons to reject the apocrypha as Christian canon:
  1. No book claims to be the inspired word of God.
  2. Some books contain serious historical inaccuracies 1, while others promote concepts contradictory to what is dominant in scripture 2.
  3. Jesus, nor the apostles ever quote from the apocrypha in the New Testament. They allude to them twice 3, but never as authoritative scripture.
  4. There are no original Hebrew copies of any apocryphal book -- the Greek is all that's available

Of the reasons to accept the apocrypha as Christian canon:
  1. Papal Decree 4
  2. Rikmeister says I was always told that the rabbinate decided that some of the Jewish litany were holy (and they became the Tanach), and some were not, and they became what is now the apocrypha. They stopped being leanrt, and therefore stopped being copied, and Hebrew editions were lost. The Catholic church disagreed with the rabbinate's decision and re-adopted the apocrypha.

The books of the Apocrypha include:

Those currently accepted by the Roman Catholic Church: Those not accepted:
Comments:

flyingroc says I recall correctly, the Septuagint, a translation of the old testament to Greek that dates before the advent of christianity included some of the deuterocanonicals, thus at least may show some jews accepted some of the "apocrypha" as canon.


Footnotes:
1: e.g. Judith and Tobit
2: e.g. Selling of indulgances, found in Ecclesiasticus 3; deifying Angels in Tobit
3: in Jude 14 and 2 Timothy 3
4: I apoligize that I can't be more objective about this. I've searched for what reasons the Catholic church gives, and this is the only one I can find. If anyone knows a better one, please /msg me.
5: These two 'books' are actually chapters of the book of Daniel, but are likely additions made many years later.
6: It's debatable whether or not to include the pseudoepigrapha as apocrypha. Many of these were written during the New Testament period as Gospels, Letters, Revelations, or Acts, but all were left out of the canon for various reasons. Most have dubious authenticity.

Works Cited:
"The Apocrypha : An American Translation" by Edgar Goodspeed (isbn 0679724524)
http://watch.pair.com/apocrypha.html
http://www.str.org/free/studies/apocryph.htm
http://www.americancatholic.org/Messenger/Sep1997/feature2.asp