Continuity mistake: In the one where Ross finds out he's having a boy (series 1), just before Rachel lets it slip, she is sitting on the edge of a chair eating a cookie. Watch her dress, it changes back and forth from being undone from the waist up, to only having two buttons open at the top.
Revealing mistake: When Emma starts crying and Rachel picks her up, look at how much the baby is moving when she takes it out of the cot. In the next shot it's suddenly only moving it's head, and very slowly at that. This is because it's a robot - and it's very obvious.
Continuity mistake: There is a long shot of Rachel and another person sitting on the sofa and Rachel's wine glass is almost empty. She then goes over to the the counter to fill it up. However, before she starts to fill it, a significant quantity of wine can be seen in the glass.
Continuity mistake: In the scene in the coffee shop where Joey tells his friends about him closing down the coffee shop when he went to the audition, as Rachel says "He left work in the middle of the day to do a personal errand?" two men walk past behind the group. As she continues talking just a few moments later, the same exact men walk past again.
Visible crew/equipment: Right after the "I'll Be There For You" intro, the camera shows the view from outside the "Central Perk" café. As it does so, a couple of stage lights are visible through the upper part of the window, obviously part of the set on the inside of the building.
Continuity mistake: When Chandler is about to put the toilet paper on the table, the picture with the little girl is facing Phoebe. In the next shot, the picture is facing Chandler. Two shots later, it's facing Phoebe again, and then Chandler again in the next shot.
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.