According to Matthew 2:1 and Luke 1:5, Jesus was born during the reign of Herod the Great, who was the Roman-appointed king of Judea from approximately 37 BC to 4 BC. Herod is the ruler who, upon hearing from the Magi that a new "King of the Jews" had been born, ordered the massacre of all male children under the age of two in Bethlehem (the "Massacre of the Innocents" in Matthew 2:16).
The other figures listed belong to different periods. Felix was a Roman procurator (governor) of Judea decades later, around AD 52-60, and appears in Acts 23-24 during Paul's imprisonment. Pontius Pilate served as prefect of Judea from approximately AD 26-36 and presided over the trial and crucifixion of Jesus as an adult. Nero was the Roman emperor from AD 54-68 and is associated with the persecution of Christians in Rome, well after Jesus' birth.
For examination purposes, remember that Herod the Great is consistently linked with the nativity narratives, while Pilate is associated with the passion (death) of Jesus.
