In Jesus' time, many Jews resented the Romans, who were occupying their country as part of the empire. This resentment gave birth to the Zealot movement, a guerrilla army which hoped to end the Roman rule.
On the other hand, some of the Jews collaborated, at least passively, with the Romans. The Pharisees collaborated in some way, looking to the future when the Romans would be gone: they would then be able to step into the power vacuum, and take control - which did in fact, happen.
There is a tense political background to the attempt of the Pharisees and Herodians to "catch Jesus in his speech." If Jesus affirmed it was lawful to pay tax to the emperor, then he would be a collaborator and would arouse the anger of many Jews. If he said no to paying the tax, he would take up the Zealot position and could be reported to the Romans for treason.
As is clear from the Gospel, Jesus had taken upon himself another way, the way of the Suffering Servant of Isaiah. He would go far beyond the political options of the day to care for the afflicted and to work for a different justice, the justice of God.
Lord Jesus, inspire us to be servants of faith and justice in union with You.