Monica lived in her own apartment for the whole series, except for a couple of episodes in season 4, where she and Rachel switched apartments with Joey and Chandler. In the series finale, they buy a house in Westchester and move there.
Chandler lives with Joey in his own apartment for the first 5 seasons, except for a couple of season 2 episodes, where he lives there with Eddie instead of Joey. 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 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 Chandler. 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 she's so hot, 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.
In seasons 1-4, Ross lives in the same apartment. In season 5, Ross moves to the apartment owned by relative of his second wife Emily, who kicks him out when Ross and Emily divorce. He spent 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 managed to procure by pretending to share Ugly Naked Guy's common interest of 'being naked', much to the dismay of the other friends who could see him across the street from Monica's apartment). This meant that all the Friends but Phoebe now lived in very close proximity.
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 gets the apartment and keeps living there. She moves in with Gary in one episode, but she moves out soon because he shot a bird. Rachel moved in with her, 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 moved in once, but he moved out soon because he didn't want to get married, ever.
Rachel never owned an apartment. For the first 5 seasons, she lives with Monica, then she moves in with Phoebe. When she has to move out because of the fire, she moves in with Joey. When she gets pregnant and Ross feels left out, she moves in with Ross. 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.