"Malicious hypocrites," Matthew calls them. They’re the Pharisees, of course; Jesus’ antagonists, and they attempt an ancient ploy: they want to present Jesus with an "either/or" decision. They want to force him into a no-win situation - darned if you do, darned if you don’t.
And so they ask: "Is it lawful to pay taxes to Caesar, or is it not?" The trap is admirably well laid. For Jesus to say "Yes" would be to antagonize his countrymen and women who staggered under the yoke of Roman oppression, who resented paying any sort of tribute or tax, no matter how trivial, to the government occupying their homeland against their will.
But to say "No" would be to antagonize the Romans themselves who cared little about First Century Palestine except for maintaining order there and collecting taxes…