Well, I need to get moving on the final book in my New Testament series. I started on James a bit back but wasn't happy with it. So to get over the hump, I'll jump to Hebrews.
The book of Hebrews is a curious book in the New Testament. It doesn't tell us its author or audience. It doesn't tell us where it was written or when. It doesn't even really tell us exactly why it was written.
Of course the reason it doesn't tell us these things is because it wasn't originally sent to us, and the author had no idea we would be reading it two thousand years later. The original author and audience knew who they were. They knew where it was being written from and where it was being written to. They knew when it was being written and what the situation was that it was addressing. We don't know these things because we are listening in on someone else's conversation.
But God of course knew we would read it later. Hebrews is for us even though it was not written to us.
To be sure, we need to be careful to distinguish between what God meant for that time and what he meant for all time. But the last two thousand years of Christian history have given us a big head start. In fact, the Spirit has developed a certain kind of "spiritual common sense" in the Church, one we probably don't even realize we have. We often apply the text directly to today in appropriate ways even when we don't know we are reading it differently than its first audiences did.
I say, "audience," because Hebrews was almost certainly meant to be read aloud to a congregation or community as a sermon rather than to be read like a book the way we read it. The author calls it a "word of exhortation" (13:22), a phrase that Acts 13:15 uses in reference to a sermon Paul preached. While Hebrews 13 looks something like the conclusion to a letter, Hebrews doesn't begin like a letter. The first twelve chapters of Hebrews look more like a sermon that the author has sent to a community in a different location than where he is. 
There is no question that Hebrews would be a lot clearer to us if we knew the answers to some of our questions about its situation. For most of church history, it was presumed that Paul was the author.  It is certainly possible, maybe even probable that the author moved in Paul's circles. But it is hard to imagine Paul saying something like Hebrews does in 2:3, where the author doesn't seem to consider himself an apostle--he only heard about salvation second hand.  In the end, we just don't know who wrote it. 
Of all the suggestions, Rome is most often suggested as the destination of this sermon, usually based on the greeting in 13:24. The idea is that there were some believers from Italy who were sending their greetings back to Rome, perhaps even Priscilla and Aquila (cf. 2 Tim. 4:19). We know enough about the church at Rome to be dangerous. While the hints of the situation in Hebrews might have applied to many places in the early church, we know enough to connect them to things we know about Rome.
Was it written to Jews or Gentiles? Was it written before or after the temple was destroyed? To many, the answers to these questions seem obvious. Surely it was written to Jews--would Gentiles understand such an intricate Jewish argument or be interested in the Jewish temple? If the temple were already destroyed, why would the author need to tell them not to rely on it?
But what seems to be common sense to us quickly unravels when we get into the mindset of believers in the first century. The earliest Gentile converts were probably God-fearers who were already worshiping in Jewish synagogues when they believed (e.g., Acts 13:16). They would have adopted the Jerusalem temple as their temple, as the temple of the living God.  At this point in history, when such Gentiles believed in Jesus, they would have seen themselves converting to a form of Judaism. The Jewish Scriptures would have become as much their Scriptures as the Bible is Scripture for us Gentile believers today. 
Similarly, Hebrews never tells the audience to stop relying on the Jerusalem temple. In fact, it never mentions the temple at all explicitly. It's argument is entirely in terms of the wilderness tabernacle that Moses constructed in the desert. Its goal is primarily a positive rather than a negative one--to place the confidence of its audience in the atonement provided by Christ. It argues for reliance on Christ rather than against worshiping in the temple. It is thus just as possible to make the case that Hebrews was written to reassure an audience in the absence of the temple as to dissuade them from utilizing it.
The problem with all these debates is that if we choose wrongly, then we have skewed our interpretation to some extent. It's probably best then, especially when we are most interested in life lessons we can take from Hebrews, to remain somewhat tentative about the original situation. It is a hard thing to pull off, but we will try...
 It is overwhelmingly likely that the author of Hebrews was a man because of the grammar of 11:32. The Greek word translated as "to tell" here is masculine singular and indicates that the author was a man.
 In fact, it is possible that the Church stopped debating whether Hebrews was good enough to be considered Scripture when most Christians came to accept Paul as its author.
 Contrast Paul in passages like 1 Corinthians 9:1 or Galatians 1:12.
 Many names have been suggested, both in the early church and in more recent times. Was it Barnabas, Luke, Clement of Rome, Apollos, even Priscilla?
 It is also clear from Acts 21:24-26 that even almost thirty years after the resurrection the believers in Jerusalem were still offering sacrifices in the Jerusalem temple.
 It is quite funny that a lot of Christians today forget that they are Gentiles. There is a strong tendency to identify so much with Israel in the Old Testament that we can almost think of ourselves as Jews (or even of America as the Israel of prophecy).