Jesus was being asked about the temple tax. The temple tax was a tax each Jew had to pay when they came to the temple. All Jews were required to come to the temple multiple times each year... so this was a direct tax on their meeting God's Commandments. Among the Jews of the time, this was a very contentious subject.
The Jews expected the Messiah to come, throw off the yoke of the Romans and reestablish the throne as a descendant of David's line. Jesus, on the other hand came as a spiritual, not political savior. As he latere stated to Pilate "My Kingdom is no part of this world".
So the tax statement by Jesus was defining a separation between 'church' and 'state'. Specifically, that his salvation, his kingdom and his followers were not there to overthrow the Romans, or protest unfair taxes. They were there to do God's Work. Thus, if Ceaser wanted the tax, they should pay the tax, so that they could focus on the spiritual, rather then being bogged down in the political.
A strict interpretation of Jesus gospel can build a strong argument that Christians should be politically neutral, focused instead on their relationship with God and their salvation through God's Kingdom.
Of course, thats assuming that the poor guy actually lived and said/did the stuff written about him. 
See, the Jews actually got a militant leader who desired the political overthrow of the Romans. Subsequent events forced Christians to strip the political and social context in order for them to survive the intense antisemitism that followed.
FYI, most of my stuff comes from this book
Cows, Pigs, Wars, and Witches. I feel it's only fair to say, but I'm going to add that I think it's pretty well backed up by other sources, one of which is a pretty reliable account contemporary with Jesus (Flavius Josephus, who was writing a defense of his people for the Romans, who were actually
there and therefore Flavius couldn't lie very much).
From what I've read, "messiah" as a concept was not a one-time thing, as the concept
at the time was. David was called a messiah, as were Saul, Solomon, priests in general, and shields, too. It might've originally meant anyone or anything that had a lot of sacred power.
Now, the vast majority Jews at the time were in a lot of trouble, as they were alienated peasants, slaves, and unemployed artisans who were living in a misruled colonial backwater and seriously fucked over by their elite (who sometimes spoke of them as if they were subhuman). There were a lot of would-be messiahs running around at the time, including both Jesus and his cousin, John (speaking of which, do you really think he was executed for his comments about Herod's marriage? No, it was for agitating too loudly against the Romans, which you can read about in Flavius Josephus's
Antiquities of the Jews).
This is pretty typical of these kind of situations even still. Having no way out of an oppressive situation + an engrossing religious community that can give you a purpose in life => religious extremism. You can see this in modern Palestine, too, in the form of suicide bombers.
Jesus was, for all intents and purposes, advocating for the overthrow of the Romans, to be replaced by a Jewish empire of the likes of David's kingdom (which is why his descent of David was so important) but on a much, much grander scale.
They would show those fucking Romans who was boss. You can see this a little when Jesus and friends attacked the temple and his comment that he was not here to bring peace, but to bring chaos.
And, just like John, he was popular enough to warrant execution so as to prevent the lower classes from rebelling.
The reason that the Gospels don't attest to his original message is because right after this, there was indeed a rebellion. A dude named Bar Kochva lead an army of 200,000 men and successfully ran an independent Jewish state for three years before the Romans managed to crush it (they killed 500,000 people and leveled a thousand villages in the process, and then sold thousands of Jews into slavery elsewhere in the empire).
Christians, particularly the gentiles Jesus' brother, James, had grudgingly allowed Paul to baptize abroad, wanted to survive this, and so as to convince the Romans that they were not like
those other, treasonous Jews, they stripped the political and social context and removed as much of the political message of Christ as they reasonably could (this accounts for a lot of the contradictions in Christ's message).
