Character mistake: When Monica and Mike are playing table tennis, Monica puts her hand on the table and looses the point. Mike then does the same thing. Surely Monica, of all people would notice that as it would have given her the point. We also see her looking at Mike at the time.
Continuity mistake: In the coffee house when Rachel is talking to Ross about trust, a couple comes in behind her and stands at the bar. In the next shot of Rachel they have switched positions; then they are gone, and you never see them walk past Chandler.
Plot hole: Chandler has always loaned Joey money for various things (rent, acting classes, Christmas gifts, etc), so he shouldn't be surprised to learn that Joey doesn't have money for fancy dinners or birthday celebrations. Plus, Joey never seems to have a problem taking money/handouts from Chandler - why now? Also, Chandler explains that they're buying Ross a cake ($20), but why doesn't Monica (the chef and his sister) just make him one?
Plot hole: This is actually for both parts of "The One in Barbados": it's shown that Monica's hair gets all frizzy from the humidity. And yet she's shown all throughout with her hair out and loose in all its frizzy glory, though it bugs her. I could maybe buy that not one of the women have hair gel with her (not bloody likely.) or that the fancy hotel doesn't sell hair gel (possible). But resourceful Monica would have thought of something to calm down her hair, or she would have just tied it up in a ponytail or bun - she does care a lot about her appearance. It makes no sense other than for comic value.
Other mistake: When Ross is helping choose a dress and shoes for Nana to wear, he is behind the chest of drawers in the cupboard. He looks in a shoebox and finds lots of pink sweet'n low packets, then pulls down another box and lots of packets fall down on him. However, it is obvious that someone is above him showering the packets down, as they are coming from directly above him, not from on top of the door.
Continuity mistake: When they are in the coffee house talking about the fact that Joey missed his audition, Phoebe says, "And that is why we don't invite you to play." Her hands are at chest height, but in the zoomed out shot, they are lowered.
Factual error: Before Ross goes out to play the second half of the rugby game, Emily gives him some tips on how to hurt the other guys. She singles out one of the guys, saying he doesn't wear a cup. These guys are all English players who love rugby. Rugby players would NEVER wear a cup for rugby is more of a grinding sport than hit and miss. These guys would be crazy to wear a cup.