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: In season 1 ("The One With George Stephanopoulos") the date's October 20th, and Ross says his birthday "was 7 months ago". In season 4 ("The One With Joey's New Girlfriend") Gunther is pretending to collect birthdays, and Ross starts saying "mine's December the..." before being cut off. In season 9 ("The One Where Emma Cries") Ross says his is on October 18th.
Revealing mistake: In the scene in the coffee house, Phoebe is talking to Monica who is drinking a cup of coffee. Towards the end of the scene, the camera is on Phoebe who is smiling & laughing at Monica. In the right hand corner is a side view of 'Monica' however, it is not Courtney Cox in the scene, but a completely different actress. [Widescreen version only, although the "wrong" nose is visible in the original].
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.
Other mistake: When Rachel arrives at Central Perk, Monica introduces her to everybody, including to Chandler, as if they don't know each other. In later episodes, we learn that Chandler met Monica and Rachel a couple of times when they were in high school (including when Monica cut his toe off). Chandler even kissed Rachel once, and since Chandler always says that he hasn't kissed many women, he should have perfectly remembered Rachel, and Monica should have remembered that the two of them knew each other.
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.
Continuity mistake: 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.