Caesarean section (or Cesarean section in American English), also known as C-section
What do you call the type of birth when the baby is not passed thru the mother's canal but cut opened and taken out?
And what do you call the regular birth?
A regular birth is called 'a normal delivery'.
And just to add to this font of information, if the first child is born C-section but the second is a normal delivery, it's called VBAC: Vaginal Birth After Cesearian. (It used to be that once a child was born C-section, all subsequent children were also C-section, but VBAC is becoming more common.)