Character mistake: During the face-off about religion/politics, they get into a shouting match about commandments, both groups being equally convinced that "honour your father and mother" is the 1st or 3rd commandment, respectively. In actual fact it's the 4th or 5th, depending on which version (Catholic or Protestant) of the 10 commandments you go by. See http://www.positiveatheism.org/crt/whichcom.htm.
Character mistake: Sam makes a joke about Laurie being "free of cataracts" because she smokes so much pot. She laughs and says "I get that. That's funny." Except there's nothing to 'get', because Sam is wrong. Cannabis has been used as pain relief for people suffering from glaucoma; it does nothing whatsoever for cataracts. It's surprising that neither Sam nor Laurie would have known this. (00:03:10)
Six Meetings Before Lunch - S1-E18
Character mistake: In this episode, the character Jeff Breckenridge meets with Josh to plan his Senate confirmation. When explaining what led him to be a civil rights lawyer, he tells how his great grandfather was captured into slavery, and "taken to New Guinea to be shipped to America". The character probably means Guinea, or Guinea Bisseau, or Equatorial Guinea, three places in the Western coast of Africa from which black slaves were sent to America. New Guinea, on the other hand, is half-the-world away, near Australia.
Character mistake: Both in the title and throughout the episode, people refer to 'Lord John Marbury'. However, the first names of peers are not used when also using their titles; so Marbury should be called either Lord Marbury (more accurately Earl Marbury), or just John Marbury. A minor error in the real world, but the White House is supposed to be aware of diplomatic eccentricities like this.
Chosen answer: A really annoying innate lack of sense. My friends and family say the same thing all the time, and I'm endlessly trying to correct them. I think people just don't know any better and (ironically) couldn't care less that they're speaking incorrectly.