When looking at Mathew 10, Jesus is sending the twelve apostles out to preach the gospel. And in this chapter He is teaching them what to do and what to expect.
I think Mathew 10:34-35 is part of this context on what to expect.
From Mathew 10:1-4 where the twelve apostles are named, and given the power to heal and cast out demons; to Mathew 10:5-15 where Jesus teaches them what to do on their journeys through the cities in Israel; all the way through to Mathew 10:40-42 where Jesus says that those who recieve the Apostles recieve Jesus who sent them and recieve God who sent Jesus. And their reward for that much is like the rewards to the apostles themselves.
From this it seems like the teaching in Mathew 10:34-35 is part of everything else regarding Jesus sending out the Apostles to teach the gospel to the cities in Israel.
Mathew 10:34-35 (NKJV)
34 “Do not think that I came to bring peace on earth. I did not come to bring peace but a sword. 35 For I have come to ‘set a man against his father, a daughter against her mother, and a daughter-in-law against her mother-in-law’;
I think in this context this is a warning for what they were going to see. There will be those who accept the gospel, and those that don't. Even within their own family. Or the people they would preach to, heal, and teach would already be in a divided state, even among their families. Enemies against eachother within the same household.
Three things this could mean (possibly all three).
1). This is who they are going to spread the gospel to, (an already divided people), or
2). this is what they will see while teaching, (preparing their hearts, this is the issues they will witness), or
3). this will be the result of their teaching. (Two for Jesus, and three against Him within the same household ).
As with everything else in the Bible I think the context is directly for the apostles, because that's who Jesus was talking to, but is also for us, because all scripture is God breathed for teaching, rebuking, and edifying. Therefore these warnings of not bringing peace but the sword is a warning to us within our own families, and warn us for those we would spread the gospel to.