Continuity mistake: When Phoebe says that they should not tell Ross that the date is cancelled so that he gets stood up, Joey says, "I hear that's bad" - implying that he has never been stood up so doesn't know how it feels. However, in TOW Two Parts (Part II), Ursula stands Joey up, and he takes it pretty bad.
Visible crew/equipment: After Jill comes to Monica's apartment, they are all around the table in the living room. As the camera is moving from Jill, to Monica and Chandler, who are sitting on the couch, a camera-man is visible in the kitchen behind them, complete with camera and trailing wires.
Continuity mistake: In the scene just before Joey goes into the shower, when he is thanking the director for the opportunity, the camera shot goes back and forth between Joey and the director several times. During this series of shots, the part in Joey's hair switches sides nearly every time. It appears that the entire shot is reversed - in the closeup, the shower curtain is on his left and the shadow to his right is sharply focused. In the lower (long) shot, the curtain is to his right, and the shadow on his left is more focused.
Visible crew/equipment: Monica is talking to Pete in the diner when he gets back from Japan. If you look at the floor at Monica's end of the counter at the very beginning of the scene, you can see the mat that is there to protect her when she falls over later.
Continuity mistake: Towards the end, during a scene at Ross and Rachel's, Sandy picks up two puppets and puts them on each hand. He starts pulling the blue one onto his left hand, but when the angle changes there's a yellow one on his left hand, and a blue one on his right. It cuts to Ross, then when we go back to Sandy they've changed hands again.
Continuity mistake: When Rachel wakes up Emma during a nap, against Phoebe's advice, there is a shot of Emma in the cot. She is wearing a plain pink hat, but in the next shot, when Rachel lifts her out of the cot, she is wearing a white hat with what looks like little flowers on it.