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.]
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.
Continuity mistake: When Monica takes off her belt and throws it onto the couch, it lands on Rachel's arm. Throughout the scene, between shots of Rachel and Monica talking, the belt constantly disappears, reappears and changes position without Rachel ever touching it.
Continuity mistake: After Marcel has escaped from the basement, he is seen eating a banana as a pair of arms reach out to grab him. These arms are supposed to belong to Mr Heckles, their weird neighbour, however the arms clearly belong to a young, large and hair free woman.
Visible crew/equipment: While Chandler is trying to break up with Janice, and Phoebe comes to help, a cameraman is visible in the background (outside of Central Perk) walking from right to left. The camera cuts away from the window angle, then cuts back, showing another cameraman walking from left to right.
Continuity mistake: Rachel takes the cat to apartment number 8 and Mr. Heckles opens the door. However, in this episode, the girls' apartment is labeled number 5 which means that number 8 is above them. In later episodes, Mr. Heckles always says that the noisy girls upstairs are keeping him from doing things. [It makes a bit more sense later on in the series when Monica and Rachel's appartment changes to number 20, which is when Mr Heckles starts to complain about the noise (although in the flash back episode when Phoebe is still living with Monica it seems apparent that they actually know Mr Heckles, which Phoebe shows no sign of when she and Rachel are looking for the owner of the cat).]