Factual error: When Leo is reminding the Congressional Aides about their bosses' drug scandals, he states that Vicodin is a Schedule II Controlled Substance. It is actually a Schedule III Controlled Substance. Given his experience in this area he'd know better.
Continuity mistake: When Doug, Toby, and Sam are talking to the President about how to handle the attempted appeal of the estate tax, they mention several times that it will be the President's first veto, and that it is something that he has never done before. However, in episode four of season two, "In this White House", in the scene where Sam is debating Ainsley Hayes on Capital Beat, the moderator asks why the President's education bill is different from the Republican proposed bill "which the President vetoed." So, the veto of the estate tax repeal was not the president's first.
Plot hole: As Josh arrives at the security gate of the White House, protestors are gathered, chanting and otherwise talking loudly. He enters the gate, makes his way to the north entrance of the West Wing, and as he enters the lobby, the crowd is still heard quite plainly, just as loudly as at the gate, as if they were right outside the door, yet the protestors are much too far away to be heard so loudly, if at all.
Continuity mistake: When Bruno and Bob are poring over exit polls, they mention that Santos seems to be leading in North Dakota, and comment that it is a state that hasn't gone Democratic in forty years. While that is true in the real world, in the fictional West Wing world, it is stated in Season 4 that Bartlet won the Dakotas in his landslide reelection.
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.
Factual error: In the scene where the Secret Service chief is in the car with Bartlet and he suddenly realizes Bartlet's injured, he yells to the driver to get to GW Hospital and the limo does a screeching U-turn on what appears to be the Arlington Memorial Bridge. At the time they were supposed to be heading from Rosslyn, VA, where the event was, back to the White House. But GW is in DC, not that far from the White House, so although the U-turn looks cool, it doesn't make geographical sense. Also, in that one shot of the outside of the car, it appears to be pouring, while back at the scene of the shooting as well as ahead of them at the hospital entrance, it's not.
Factual error: Sam says the "state-of-the-art" oil tanker he recommends can carry 2.2 million gallons of oil and weighs 308,000 tons. The Exxon Valdez could carry over 60 million gallons (1.48 million barrels) and weighed only 211,500 tons. The writers mistakenly swapped 'gallons' for 'barrels' so that this enormous tanker carries very little oil. It would actually carry 2.2 million "barrels", which equals 92 million gallons.
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").
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.
Plot hole: The Zoey Bartlett kidnap crisis cliffhanger that stretches across the end of Season 4 and the beginning of Season 5 makes no sense in terms of time. Charlie pulls out a note from his wallet that says they will dig up a bottle of champagne on May 7 - Zoey's graduation night. They do. Later that night she is abducted. She is gone three days. Four days later (in "Jefferson Lives") Abby chews out Leo, saying, "It's only been four days and her bruises have not yet begun to heal." The date should be May 14th. However, "Jefferson Lives" supposedly occurs on the Fourth of July - Zoey says it's the Fourth and they all watch the Capitol Fireworks at the end. Boy, that sure was a short month of June.
Continuity mistake: Bartlett is in the executive office in the residence watching four TVs at once. He hears a line in the black and white movie, a second later he rewinds it to hear the line again - he rewinds way too much and for too long, but once he stops the tape, it plays the line he wanted to hear again right away. He went too far back (for visual effect) than what in reality it would have taken.