Jesus was born in a manger which may have been in a cave or a shed, in Bethlehem. The events surrounding the birth are called the Nativity, but the whole birth story starts with an earlier event when the Angel Gabriel told Mary that Jesus would be born to her. That first event is called the Annunciation.
The days leading up to the birth are known as Advent. Advent starts on the fourth Sunday before December 25.
The celebration of the birth is called Christmas.