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)
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.
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.