What is the relationship between Pilate and Herod?

Apparently not much of a relationship.

Pilate was Governor of Judea. Herod Antipas was district ruler of neighboring Galilee. (There were many other in the line of Herod's."

Actually Pilate and Herod had been enemies, possibly because of certain accusations that Herod had leveled against Pilate.

But on the last day of Jesus' earthly life, when Jesus was brought before Pilate; Pilate had heard that Jesus was a Galilean. Herod was visiting in Jerusalem at the time; so Pilate sent Jesus to Herod Antipas, for Pilate had experienced trouble with the Galileans.

This move on Pilate's part pleased Herod and they became friends. Acts 4:27 connects the two rulers as colluding against Jesus.