Character mistake: George says that he's paying the chiropractor only half of his $75 bill. However at the end episode Jerry tells George that he gave the chiropractor $35. (00:12:50 - 00:19:20)
The Robbery - S1-E2
Character mistake: When Jerry is selling his couch to Elaine, he tells her to take it out of the door and leave - but since the arrangement is that she would move to his place, that does not make sense. (00:12:10)
The Robbery - S1-E2
Character mistake: At the end of the episode, the waitress introduces her next door neighbour, a hot redhead. After saying her line, she leaves, and George says "Nice to meet you..." and is stuck shaking hands with Carol, who he already met before and invited him to the party to begin with. (00:21:45)
Continuity mistake: George drinks from the bottle of anti-acid medicine, puts the lid back on, then at the "Yes, she wants me to like her" bit the lid is off again. (00:08:40)
Other mistake: Jerry is at the party Elaine's friends have during the marathon. The clock in their living room never changes the displayed hour. (00:15:20 - 00:18:50)
Revealing mistake: During the shopping scene with Vanessa, the clock on the wall can be seen for over a minute but the hands never move. (00:05:30 - 00:06:40)
Continuity mistake: In the pilot episode "Good News, Bad News", look out when the waitress discusses the two coffee pots. She says "Trusts me George, nobody wants to see you on caffeine", and the unbuttoned top of Jerry's jumper is noticeably folded - but straightens instantly in his reaction shot. Also it should be noted that she walks away without actually pouring coffee to George - you have to assume a little amount of time elapsed between that reaction shot with Jerry and the following on George. (00:01:45)
Factual error: When Peterman is making a toast at his party, he quotes what he says is something said by the Papa New Guineans, and he proceeds to pronounce a sentence filled with clicks (lingual-ingressive consonants). However, this is impossible, as clicks are only used in southern African languages and three east African languages. (00:03:10)
Revealing mistake: When Jerry is slamming the payphone in frustration with Elaine, he slams it on The Switch-hook multiple times. Pressing down The Switch-hook like that would end the call. Clearly Jerry is not really connected to Elaine on the phone. (00:14:05)
Continuity mistake: When Elaine gets kicked out of the nail salon it is a clear day (in fact the whole episode is) then the next scene she meets J Peterman at night and in the rain. When she left the salon she is wearing a red blouse, no jacket. When she meets Peterman she is in a trench coat with a white blouse. The Peterman portion of this episode really makes no sense.
The Implant - S4-E19
Continuity mistake: During the wake for Betsy's aunt, George puts snacks on his plate while talking to the reverend. He puts at least 6 pieces on his plate while the next shot of his plate shows only 2. (00:14:15)
Continuity mistake: When Kramer enters Jerry's apartment, he closes the door. In the next shot, the door is wide open. A few shots later, George opens the door, which was closed when he entered.
The Ticket - S4-E4
Continuity mistake: George is in Jerry's apartment and he is unshaved, maybe a 2-day beard. However in the next scene after he leaves The Apartment with Jerry, he is shaved again.
Continuity mistake: When George enters The Cafe near the start, he takes Jerry's plate and begins eating the fries. After finishing the first fry, his hands are on the plate. Jerry then asks him what happened, and suddenly he has a fry in his hands which are also now raised slightly.
Factual error: With the size of that red ball (approximately 2.5 foot diameter) and the density of oil (approximately 60 lb/ft^3), the giant ball of oil would weigh 491 pounds. This would be impossible for Kramer and Darren to lift out of the window and would definitely kill anyone it beaned.
The Apology - S9-E9
Continuity mistake: When Jerry and his naked girlfriend are playing scrabble just after Jerry accuses her of stealing and it flips to the front at the bottom of the screen you can see a bra or towel covering her breasts but is gone when she stands up. (00:03:40)
The Airport - S4-E12
Deliberate mistake: In this episode Elaine angers the bag man and he puts her bag on a plane to Honolulu. When Jerry and Elaine arrive in New York and are waiting for their bags, it cuts to the "Hawaii" airport and shows Elaine's bag coming out to the baggage claim belt. This was a deliberate mistake as there is no way Elaine's bag could have gotten to Hawaii at the same time Jerry and Elaine are waiting for their bags in New York, since they only left from Missouri.
Chosen answer: George does mention his brother in the episode "The Parking Space". They actually really never mention anything that contradicts the fact he had a brother though, it just isn't mentioned.