Date: There has been a lot of debate over when Jude was written. Some have argued for a date between the late first century and the end of the second century. Some conversely argue for a date as early as the 50s AD. A late date is usually suggested for the following reasons:
It is suggested by some that Jude is combating Gnosticism, which was a second century heresy that denied the humanity of Christ. I think that suggestion is incorrect. There is no clue in the letter that the infiltrators misunderstood Jesus nature as both God and man. One would expect that if they were advancing a teaching that gravely erroneous that Jude would directly address that in his letter. The antinomianism that Jude addresses, while being characteristic of gnostic 'Christianity,' was a problem long before Gnosticism arrived on the scene.
In Jude 3, Jude writes to urge them, 'to contend for the faith that the Lord has once for all entrusted to us' (TNIV). In verse 17 they are told to, 'remember what the apostles of our Lord Jesus Christ foretold' (TNIV). Some have suggested that these two verses give the impression that Jude was written in a period after the death of the apostles where standardization of church teaching has occurred. All these verses imply, though, is that there is no apostolic presence any longer in the locale of the recipients. They received the contents of the gospel from the apostolic founders of their church, and they are to hold fast to their teaching.
As we have seen, a late date is not required by the internal evidence of the letter. My view on authorship below commits me to a date in the 50s or 60s.
Authorship: It was clearly written by a Jew. Old Testament allusions are translations from Hebrew (they do not follow the wording and even sometimes the meaning of the LXX). The internal witness of the book is that it was written by Jude brother of James. While there were several Judes (Judases) mentioned in the NT, only one is mentioned as a brother of James, that is Jude, brother of Jesus. James was also a common name at the time, but only one James was well known enough to go by, simply, James. That was James brother of Jesus, leader of the Jerusalem church. Both of these factors lead to an identification of Jude as Jude brother of Jesus.
Some have suggested, though, that Jude was a pseudonym for some later author, either writing in Jude's honor/memory, or piggybacking on his authority. Usually this approach is used when the contents of the letter could not have been written during the time frame when the author lived. However, as we saw above, nothing in the letter would have been inapplicable to the 50s or 60s when Jude lived. Also, Jude was too obscure of a figure to be a likely choice of pseudonym.
Audience: Who was Jude written to? Clearly it was written to a Jewish audience. It's not reasonable to assume that a Gentile audience would be able to identify the multiple allusions to Jewish intertestamental literature (1 Enoch and The Assumption of Moses). Some have suggested Syria, but I find that unlikely since Jude was not accepted as canonical by the Syrian church. Palestine has also been suggested, but one would expect substantial apostolic activity in Palestine when Jude was written. Also, Aramaic would have been the likely language used for a letter from Jews to Jews. A third suggestion is Egypt. Not only did Egypt have a large Jewish Christian population, but the book was accepted as canonical very early there. Also, apocalyptic literature like 1 Enoch was widely read by their Jewish and Jewish Christian communities. This makes Egypt the most plausible destination of the letter in my opinion.