Monica
Monica lives in her own apartment for the whole series, except for a couple of episodes in season 4, where she and Rachel switch apartments with Joey and Chandler. Starting from the beginning of season 6, Chandler lives with Monica in her apartment. In the series finale, they buy a house in Westchester and move there.
Chandler
Chandler lives with Joey in his own apartment for the first five seasons, except for a couple of season 2 episodes, where he lives there with Eddie instead of Joey and for a couple of episodes in Season 4 where he and Joey switch apartments with the girls. In the beginning of season 6, he moves in with Monica in her apartment. In the series finale, they buy a house in Westchester and move there.
Joey
Joey lives in the same apartment for the whole series, except for a couple of episodes in season 2. He thinks it's a good idea to live by himself, but when he is fired and he misses Chandler, he moves back in with him. He also lives in Monica's apartment with Chandler after the girls lose a bet in "The One With The Embryos". After Chandler moves out, he takes Janine as a roommate because he is attracted to her, and eventually they start a relationship. When they break up, he starts living alone. From season 7 onwards to the end of the series, with the exception of season 8, Rachel lives with him.
Ross
In seasons 1-4, Ross lives in the same apartment. In season 5, Ross moves to the apartment owned by a relative of his second wife Emily, who kicks him out when they divorce. He spends a short time living in Chandler and Joey's apartment before settling in the apartment directly opposite Monica's, which had once been inhabited by Ugly Naked Guy (which he manages to procure by pretending to share Ugly Naked Guy's common interest of 'being naked', much to the dismay of the other friends who can see him across the street from Monica's apartment). This means that all the Friends except for Phoebe now live in very close proximity.
Phoebe
Phoebe lives in the same apartment for whole the series, mainly alone. In the beginning of the series, she lives with her grandmother, Frances, who also owns the apartment. When she dies, Phoebe inherits the apartment and keeps living there. She moves in with Gary in one episode, but she leaves soon after because he shoots a bird. Rachel lives with her for a while, but when the apartment burns down she has to move out. In the rebuilt apartment there's only one bedroom, so Rachel can't move back in. Mike also moves in with Phoebe once, but he leaves soon after because he doesn't want to get married. However, he moves in again when the couple gets back together in season 9.
Rachel
Rachel never owns an apartment. For the first five seasons, she lives with Monica before she moves in with Phoebe. When she has to move out because of the fire, she stays with Joey. When she gets pregnant, she moves in with Ross so that he can support her and the baby. After the birth of Emma, she moves out because of a fight and lives with Joey for the rest of the show. After the series finale, she presumably moves back in with Ross.