The One With All The Haste - S4-E19
Factual error: The guys are gone for a single night, watching the Knicks game. (You can tell this because they talk about how "we can go see the game tonight," and when they get back everyone's wearing the same clothes as that conversation.) When they get back, both apartments are completely switched back: the large, heavy furniture is replaced, the posters are re-hung, etc. Even if they were gone from, say, 6 p.m. to midnight or so, there is no way that Rachel and Monica would be able to make that transition in that kind of time.
The One with the Fake Monica - S1-E21
Factual error: The monkey is allowed outside his cage at an airport gate. Animals, with the exception of assistance dogs, cannot be transported in the cabin with passengers, and this rule applied in the 1990s as well. There is no way a monkey would be brought through a security checkpoint into a terminal concourse of a NYC airport and allowed aboard a regular commercial airliner.
The One Where Everybody Finds Out - S5-E14
Continuity mistake: In the scene where Chandler turns on music when he and Phoebe are going to "have all the sex" you see Phoebe's arm begin to move in the right corner of the screen. When it cuts to her she has not yet begun dancing, and when he goes to turn the music on again it is a duplicate clip from the initial shot.

The One Where The Stripper Cries - S10-E11
Continuity mistake: When Ross and Chandler are talking about who broke which pact in college (the girls they agreed not to date) they are standing at the drinks table. Chandler is holding a drink in his left hand and his right hand is in his pants pocket. The shot then changes to face Ross, Chandler does not remove his hand from his pocket, but in the next shot facing him his drink is in his right hand. He then places it on the table. (00:18:20)

The One With The Cake - S10-E4
Continuity mistake: When Monica's mum and dad are leaving the party, they are standing next to each other while her dad explains why he can not drive anymore. Her mum has both hands in front of her holding on to her purse, in the next shot one arm is around Monica. (00:13:50)
The One With The Embryos - S4-E12
Character mistake: When they are about to start the game, Ross declares that "the first team to answer the most questions wins." This is a nonsensical statement. By that logic, the first team to answer a question correctly would be the winner. It should be either the team that has correctly answered the most questions at the end of the game wins, or the first team to correctly answer an X number of questions.
Suggested correction: While it's a weird way for him to say it, he starts by saying "each team will answer ten questions", so no, the first team to answer a question wouldn't win.
Yes, he qualified that it was a 10-question game, and we as the audience understand going in how the winner is decided. It's just the way he declared how the winner would be decided didn't make sense. It would be like watching an NBA game, and before tipoff, the crew chief tells both teams that the first team to score the most baskets wins. We know what the rules of an NBA game are, but the referee makes a nonsensical statement all the same.
Except there's so many variations of how a team can win a game; best 3 out of 5, win by 2, or a game can even end in a tie. Since this isn't a professional sport where all the players know how a game ends, Ross is basically saying after 10 rounds, if there is a tie, it will go to sudden death with the team with the first correct answer winning, just not in so many words.
The One with the Blackout - S1-E7
Other mistake: Ross says that he was at Disneyland with Carol and got caught hooking up on Small World and asked to "never to return to the Magic Kingdom." Disneyland is in California and the Magic Kingdom is part of Disney World in Florida. (00:06:10)
The One Where Ross Got High - S6-E9
Plot hole: Monica is trusting Rachel to make a traditional English trifle and being a top chef (or an even averagely intelligent human being) she would know what ingredients are used in one. So there's no way she could have missed Rachel cooking beef with peas and onions during the cooking process. Rachel's cooking it right there in Monica's kitchen while everyone else is there!
The One Where Rachel's Sister Babysits - S10-E5
Other mistake: Jennifer Aniston's tan mark from her wedding ring is especially visible in this episode. Most notably when she says "But Erin Brockovich had her own house."
The One Where Ross And Rachel Take A Break - S3-E15
Revealing mistake: In the scene where Ross brings the picnic basket to Rachel's office, he lights a candle and puts it near a plant (which then sets it on fire). However, the candle is barely (if at all) close enough to set it alight, and when it does, a huge flame just erupts from the middle of the plant, not the edge where the candle was.
The One With Chandler In A Box - S4-E8
Visible crew/equipment: When Chandler is climbing out of the box, you can see the bottom of the set spotlight right above the bathroom door. (00:20:35)
The One With The Memorial Service - S9-E17
Continuity mistake: After Phoebe kisses Mike in her apartment, Mikes friend rushes in and discovers them. As soon as the shot widens, Mike's hand moves from being behind Phoebes head to his lap. (00:17:35)
The One With Phoebe's Wedding - S10-E12
Revealing mistake: Just before we see everyone setting the wedding up outside, there is a sweep shot of the city with snow falling. It's very obvious that this snow is added digitally and it's very badly done. There is no snow on any of the buildings, and there is no snow hitting the surface of the water. (00:20:50)
The One After Joey And Rachel Kiss - S10-E1
Continuity mistake: When Ross and Charlie are sitting on the bed it shows Charlie fixing her T-shirt. In the very next shot her hands are on her lap. (00:04:25)
The One With Phoebe's Wedding - S10-E12
Revealing mistake: At the wedding, none of the snow melts when it has been sitting on people for several minutes. Occasionally when it gets brushed off clothes it just sticks to hands and sleeves, again for several minutes without melting. (00:21:15)

The One With Ross's Tan - S10-E3
Continuity mistake: In the scene where Joey and Rachel are sitting on the couch in their apartment after they decide their relationship is hopeless Rachel puts her head on Joey's shoulder. Her hair is on her face but when she lifts her head up in the very next shot her hair is now behind her right ear which she couldn't have done because her hands never left her lap. (00:22:00)
The One With Phoebe's Rats - S9-E12
Continuity mistake: When Monica goes to Rachel's office and invites Gavin to Rachel's birthday party Rachel gets up and goes to the door exclaiming "why did you invite him, I hate that guy." She has nothing in her hand when she gets up from her desk; but at the very end of the scene when she tells Monica to be quiet, she's got a pen in her hand for one shot. (00:12:45)

The One With Phoebe's Rats - S9-E12
Continuity mistake: At the party, when Monica and Rachel are talking about Gavin, Rachel has a napkin in her hand at one shot, the shot changes and the napkin is replaced by a drink. (00:13:20)
The One Where Monica Sings - S9-E13
Revealing mistake: When Joey has his eyebrows plucked at the salon, it is obvious that the woman is not pulling any hair off. (00:07:00)

The One Where The Stripper Cries - S10-E11
Continuity mistake: When the girls and Mr Goodbody are talking about paying him the three hundred dollars, the top buttons of his shirt are ripped open. Half way though the scene this changes to only one button being open and his shirt looking neat and pressed, as if he had not started his strip yet. (00:13:25)