What is it called when you have two groups and one starts singing a song and part way through the second group begins singing the same song?

This is called a round. A common round is "Three Blind Mice"

Three blind mice

Three blind mice

See how they run

See how they run

They all run after the farmer's wife

She cut off their tails with a carving knife

Have you ever seen such a thing in your life

as three blind mice.

Normally the second group of singers start the song from the beginning after the first group are about to sing the second line. In this song the second line has the same tune as the first line so you don't really hear the effect of the round until the third line.

As the first groups sings "See how they run" the second group is singing the second line "Three blind mice" You start to hear harmonies at this point which make the song more interesting.