Continuity mistake: In the one with Phoebe's rat babies, it is very obvious that the box has nothing in it. Also when she is in her apartment explaining about giving up the triplets she had for her brother she puts the box down on the table, yet when it cuts to Mike's face and Phoebe's realisation that she hadn't told him that before, the box is back on her lap.
Continuity mistake: When Rachel finds out that Ross and Julie are getting a cat, watch Rachel's neck. When she says "together?" she's not wearing a necklace; cut to Ross and Julie for a fraction of a second, then when it cuts back to Rachel saying "both of you?" she's suddenly wearing a necklace. It then disappears again when she says "isn't that just lovely?"
Continuity mistake: When Joey is counting the floors of Ross' building (the first time) to find out where Hot Girl lives, he is wearing a black top with a zip. When he rushes over to Ross' building he walks down the hall and knocks on what he believes to be Hot Girl's door. In these shots he is wearing a dark purple button up shirt. As soon as Ross answers the door, Joey is wearing the black top with a zip again.
Continuity mistake: The number of the flats near the start of the first series was 4 and 5. After a few shows they changed to 19 and 20. [The producers realised that the numbers 4 and 5 were for apartments on the lower floors and had shown Monica's apartment to be much higher up so they changed the numbers - it's still a mistake but there's the reason.]
Continuity mistake: When Joey goes to Monica and Chandler's apartment, he goes to the fridge to get something to drink to keep his bladder full for the audition. When he closes the fridge he has two bottles in his left hand. The camera cuts to Monica, then cuts back to Joey. He then has two cans in his left hand and is holding them differently.
Visible crew/equipment: Chandler and Joey are spending time with Richard; as they play foosball and Richard scores, he leans back and says "Could that shot BE any prettier?" in a mockery of Chandler. The end of the wall and the hallway behind it are visible (ie the wall set ends).
Continuity mistake: Monica is sitting in the living room with all her wedding presents, waiting for Chandler to return. She begins to open a large parcel and reveals a brown cardboard box. Then the camera moves to Joey, and when it moves back, the parcel is wrapped up again. This happens about three times.
Plot hole: During this episode, Chandler is having major problems slipping people dollar bills. He asks Monica if Richard (her ex) use to slip people notes. But in Series 2 there was an episode where Chandler and Joey were doing impressions of Richard; they impersonate him by slipping money to each other with no problems. So not only has Chandler forgotten that Richard used to slip money, he's also lost the ability to slip money himself.
Continuity mistake: When Monica is trying to cheer Joey up, she tells him that she straightened out his shower curtain so it wouldn't get mildew. But in a previous episode (The One Where Eddie Moves In), Joey gives the gang a tour of his new apartment; the shower is clearly visible in the bathroom, and it has a door.