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.
Continuity mistake: While Mama Joey is looking for Chick Jr. and Duck Jr., the things pinned to the bulletin board beside the door have vanished when Monica and Chandler walk in, but after Monica walks out to check on the twins everything's back up on the board like it was before.
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.
Plot hole: When Chandler is in Joey's apartment, he answers the phone and says, "Joey's not here right now, can I take a message?" A split-second later, we hear Chandler say, "Yeah, so the audition's been moved from 5:00 to 2:30?" It would be 100% impossible, within the fraction of a second that Chandler is silent, for the other party to pass along the fact that Joey's audition has been moved and the new start time. In fact, there wouldn't have been time to speak more than a couple of syllables.
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.
Audio problem: The camera cuts briefly to Chandler as the friends are discussing Monica's love life. During this very brief time, you can see Phoebe's mouth moving but there is no audio of her voice, and Monica's voice is heard talking in the background instead.