Deliberate mistake: This show is famous / infamous for having many long conversations between characters that are walking the corridors of the West Wing of the White House. In this episode, the chat was long enough that more space was needed to complete the walk, so the actors were instantaneously teleported to a different spot to lengthen the walk. Using http://www.whitehousemuseum.org/special/wwtv.htm as a guide, Pres. Bartlet and Leo leave the Oval Office, pass through the Presidential Secretary's office, turn left, passing between the Roosevelt Room and the Chief of Staff's office. At the end of the Roosevelt room, they turn left through dark wooden doors that are open. Instantly, the actors are at the next junction higher on the map (no wood doors), as if they'd just passed between the Roosevelt room and the Communications Bullpen, and they continue (downward on the map) through the intersection where they original turned (watch for the doors after Leo says "You wouldn't understand"). (00:02:40)
Continuity mistake: As Jed and Leo pass through a doorway, the left door is open toward the camera while the right door is opened away. But as the camera angle shifts, the left is opened the same as the right. (00:03:25)
Revealing mistake: Oliver Babish's office has windows in a location it can not possibly have them. Jed and Leo are in continuous conversation as they leave the 'normal' West Wing set at the end of a hall (See http://www.whitehousemuseum.org/special/wwtv.htm where the words 'Press Briefing Room' are - that's where they turn to their right). They walk a few steps then turn right again and ahead of them is Babish's office, with windows straight ahead. If this had been a single continuous set, Babish's office would be located in the stair area to the left of the press briefing room, and the windows would offer a view of the hallway that runs between the lower corners of Leo and C.J.'s offices (seen at 3:36), not the outside world. (00:03:30)
Character mistake: Someone set up the chess board incorrectly. White should be on the right. (00:09:20)
Character mistake: When Josh is giving his WWII example to Donna, he talks about "France, Austria and England being pounded by the Germans." Except Austria was incorporated into the Third Reich after the Anschluss of 1938; it was essentially part of Germany for all intents and purposes. It therefore was not being attacked by the Germans at all. One would have thought Josh, who shows a firm grasp of WWII information in other episodes, would have known this. (00:31:20)
Visible crew/equipment: As Bartlet enters the Oval Office, Babish stands waiting inside. The shadow of the panning camera darkens Babish's face considerably as it moves right to left. (00:39:20)
Plot hole: In Season 1, Episodes 5&6, it's mentioned several times that Zoey is already 19 a few weeks before she starts college. Yet in this episode, Charlie points out to Leo and the President later tells Oliver Babbish that Zoey was 17 when she was filling out her enrollment forms for college, therefore a parent had to sign them. These are enrollment forms, not application forms - it's extremely unlikely that they were filled out over a year before she started school.
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.