Continuity mistake: When Ross gives the super the money and asks "will this help with the knob" the super says that the place isn't open until Tuesday. When he says that the money is in his right hand. Then when he asks "am I not saying it right", the money in his right hand is gone.
Continuity mistake: When Mike says "I am going to make this decision for us. What do you think we should do?" he is holding the check. The very next shot shows the charity worker taking the check off the desk and giving it back even though Mike never put it down.
Character mistake: Ross tells Chandler to run a "post pattern to the left." (And Chandler quickly agrees.) In football, this is a type of passing play where Ross would hold on to the ball while Chandler runs straight down the field immediately after the snap, eventually veering diagonally to the left once he's a ways down field to receive a pass from Ross. The play that they actually run is completely different. It's a running play known as a sweep. (A toss sweep to be precise.) There is no possible way that anyone either making play calls or running them would ever confuse a post vs. a sweep.
Deliberate mistake: The door to Monica's bedroom is open in many shots, such as when Ross introduces Julie, and we can see that there's a wall (with a framed picture) directly opposite the doorway, but when Phoebe's giving Monica the haircut they're in Monica's bedroom and there's no such wall in front of the door.
Other mistake: After Rachel takes the phone from Ross and hangs up on Julie, when she's sitting at the table there's a bottle of Evian water in front of Monica with part of its brand name covered - the letters "e" and "n" are whited out leaving "via" on the bottle.
Factual error: Phoebe holds up a bag of quarters that is almost the size of her head. This would be physically impossible for anyone except perhaps the world's strongest weight-lifters. Quarters are exceptionally heavy for their size. Two rolls weigh exactly a pound. It would take at least 300-400 rolls worth of quarters to fill up the bag Phoebe is holding, which means she is lifting 150-200+ pounds, primarily with one arm.
Continuity mistake: As Chandler is writing on the board to remind himself about Joey's audition, the door opens as he's writing "Joey", and there are a bunch of squiggly lines under the y. As Chandler is thinking back at that moment and it moves to present tense, he looks over to the board, and the squiggly lines are different.
Continuity mistake: When they're all sitting together and having pizza, talking about parents, Rachel has no napkin. When she says: "Just think, in a couple of years we get to turn into them", a bunched up napkin appears beside her on the couch. After Chandler's remark it has disappeared again, then it appears and reappears again between shots.
Continuity mistake: When Monica asks Chandler and Joey to take Richard to the Knicks, Joey tells Monica, "We will bring him, but only if he takes the Jaguar." Chandler then drops his head in mortification, and the scene then cuts to focus on Monica's expression. In the right corner of this cut, you can see Chandler is looking ahead, and his mouth is moving as if he is continuing a conversation, but there is nothing being said during this scene. The camera then cuts to a different angle where Chandler lifts his head from the earlier drop just before he speaks to Joey.