King Herod the Great (or Herod I) was the king when Jesus was born. Mathew 2:1-19. He was not the only king in his family to use the name "Herod".
He died:
Mathew 2:14-15
So he got up, took the child and his mother during the night and left for Egypt, where he stayed until the death of Herod. And so was fulfilled what the Lord had said through the prophet: "Out of Egypt I called my son."
Mathew 2:19
After Herod died, an angel of the Lord appeared in a dream to Joseph in Egypt 20and said, "Get up, take the child and his mother and go to the land of Israel, for those who were trying to take the child's life are dead."
His successor in Judea was his son Archelaus or Herod Archelaus.
Mathew 2:21 - 23
So he got up, took the child and his mother and went to the land of Israel. But when he heard that Archelaus was reigning in Judea in place of his father Herod, he was afraid to go there. Having been warned in a dream, he withdrew to the district of Galilee, and he went and lived in a town called Nazareth. So was fulfilled what was said through the prophets: "He will be called a Nazarene."
There is also another son of Herod the Great named Herod Antipas who was made a tetrarca (ruler) in Galilea. This Herod Antipas is the one who apperas during the adult times of John the Baptist and Jesus.
Mathew 14:1-3
At that time Herod the tetrarch heard the reports about Jesus, and he said to his attendants, "This is John the Baptist; he has risen from the dead! That is why miraculous powers are at work in him."
Now Herod had arrested John and bound him and put him in prison because of Herodias, his brother Philip's wife,
The King Herod in Acts 12 is Herod Agrippa,nephew of Antipas, grandson of Herod the Great.