The dancer on the left (usually a boy) will place his right hand on the back of the other dancer (usually a girl). The dancer on the left will be holding his left hand out in front of them, with his palm up, while the dancer on the right puts her left hand on it, palm down. So they move as if the boy is helping the girl turn by guiding/pushing her, but it should be a very gentle "push". In general, in square dancing, each dancer is responsible for his/her own action and should be providing the bulk of the energy needed to move from place to place using his/her own feet.
Unless the caller adds more words to indicate otherwise, the turning ends when the couple is facing another couple, or the center of the set. This will most commonly be after turning 1/2 (180 degrees), but depending on the positions of the couples when they start it might be some other amount.