What is the Apocrypha and what's it doing in my Bible?

The Apocrypha is a set of books written between 400 BC and the time of Jesus' ministry on Earth. These books consist of 1 and 2 Esdras, Tobit, Judith, the Rest of Esther, the Wisdom of Solomon, Ecclesiasticus (also known as Sirach), Baruch, The Letter of Jeremiah, Song of the Three Young Men, Susanna, Bel and the Dragon, The Additions to Daniel, The Prayer of Manasseh, and 1 and 2 Maccabees.

Protestants reject the apocrypha as being inspired for one very good reason - the creators of the Old Testament, the Jewish community, rejects the apocrypha. So why does it exist in some bibles? 

To be honest an easier question to answer is "Why isn't the Apocrypha in most bibles?" As I mentioned before these books were never acknowledged as sacred Scriptures by the Jewish Church. Reflecting this Martin Luther removed them from his bibles at the beginning of the reformation. The Manual of Discipline in the Dead Sea Scrolls rejected the apocrypha as inspired. Early church fathers seemed to line up to reject the apocrypha; Melito of Sardis (AD 170), Origen Adamantius, Eusebius of Caesarea, Cyril of Jerusalem, Hilary of Pointiers all rejected the apocrypha. Even ancient historian Titus Flavius Josephus (Josephus of Antiquities) rejected the apocryphal books 

"From Artexerxes to our own time the complete history has been written but has not been deemed worthy of equal credit with the earlier records because of the failure of the exact succession of the prophets." ... "We have not an innumerable multitude of books among us, disagreeing from and contradicting one another, but only twenty-two books, which contain the records of all the past times; which are justly believed to be divine..."(Flavius Josephus, Against Apion 1:8) 

The books of the apocrypha were written in the silent years, the time between the last prophet and Jesus ministry on earth. In about 435 BC the prophet Malachi ceased writing and the voice of God was no longer heard in the land through His prophets. Israel had returned from captivity but were a puppet state, Zerubbabel was the crown prince and heir to the throne of David but he remained a governor for the Babylonian government, ruling Israel as a vassal state. Greece rose and overthrew Babylon, and eventually Rome rose and conquered Greece taking all of Greece's holding in the middle east. It was in this period that the apocryphal books were written, a time when no prophet of God spoke. Jesus Himself referred to the entire Jewish cannon but never mentioned the writings in the apocrypha. In fact the apocrypha was never ever mentioned in the New Testament. 

The teaching of the apocrypha is also very different from the bible. Where God demanded harmony between the sexes, the apocrypha seems to promote disharmony:

19 Any evil is small when compared to a woman's evil; may she experience a sinner's fate. 20 A talkative wife for a quiet husband is like a sand dune beneath the feet of an elderly person. 21 Don't fall for a woman's beauty, and don't long for a woman. 22 There will be anger, shamelessness, and utter disgrace when a wife provides for her husband. 23 A wicked wife causes a humiliated heart, a gloomy face, and a wounded heart. Drooping hands and weakened knees come from a woman who doesn't make her husband happy. 24 Sin began with a woman, and because of her all of us die. 25 Don't allow an outlet for water, and don't give a wicked wife freedom to speak. 26 If she doesn't do as you say, divorce her. (Sirach 25:19-26)

Other ungodly behavior is promoted in the apocrypha, the command by an angel to use magic (Tobit 6:5-7), that almsgiving results in the forgiveness of sin (Tobit 12:9) and the offering of money for the forgiveness of sins of the dead (2 Maccabees 12:43-45)

So why is the apocrypha in some bibles? Because the Roman Catholic Church wants it there. Eusebius Hieronymus Sophronius, or Jerome to his friends, was secretary to Pope Damasus in 382 and was probably the greatest Christian scholar in the world. In Jerome's day the entire bible was available in Greek, the common language of the eastern Roman provinces for over 700 years, but the west spoke Latin so Pope Damasus charged Jerome with the task of translating the Greek bible to Latin, which became the Latin Vulgate. When translating the Old Testament Jerome started with the Greek translation, the Septuagint, but then realized to get an accurate translation he would have to go to the Hebrew source documentation.

As Jerome translated the Old Testament he came to the realization that the books that the Jews considered Holy Scripture did not include the apocryphal books. Jerome was compelled by the church to include them. But he made it clear that in his opinion the Apocryphal books were only liber ecclesiastici (church books to be read for edification), as opposed to the fully inspired liber canonici (canonical books to establish doctrine). Over 1,000 years later the Protestant Reformation would remove these books from the bible.

But the Roman Catholic Church chooses to keep the apocrypha because it supports several unbiblical doctrines that the Roman Catholic Church adores, such as almsgiving for the forgiveness of sin A.K.A.indulgences (Tobit 12:9) and the offering of money for the forgiveness of sins of the dead (this is also used as a reference to the heresy of purgatory)(2 Maccabees 12:43-45), prayers for the dead (2 Maccabees 12:44), the expiatory sacrifice which eventually became the Mass (2 Maccabees 12:39-46), the worship of angels (Tobit 12:12), invocation and intercession of the saints (2 Maccabees 15:14; Baruch 3:4)

No Roman Catholic was required to believe that the Apocrypha was scripture until 1546 when the Council of Trent declared them scripture. Prior to the Council of Trent, the Apocrypha were read and used, but they were always classified as Apocrypha (meaning secret or hidden), as Jerome called them, "doubtful writings." The Council of Trent ordered the removal of the idea of "doubtful" and unilaterally declared them to be inspired by God. This is one of the reasons why 1546 is marked as a major change in Roman Catholic belief. Two prior councils: the Council of Laodicea in AD 367 and the Fourth General Council Chalcedon in AD 451 both gave lists of the recognized books of the Bible and neither one included the Apocrypha.

So in a nutshell the main reason why Roman Catholics receive the apocryphal books as Scripture is because the Roman Catholic Church says so. The Roman Catholic Church believes that the Roman Catholic Church is the final authority on all matters of faith and practice whatever it says is final regardless of what God says and the matter is not up for debate.