Continuity mistake: Leo enters the situation room and is told the "planes are on the deck". He watches a video feed as a plane is launched from the USS Lincoln. The S3-B Viking is seen approaching one camera (single vertical tail fin, wing-mounted engines), then from another angle, we see an F-14 Tomcat lifting off (dual tails and fuselage-mounted engines) and moving away from camera. This is supposed to be live video, and it isn't possible two different jets are launching from the same carrier at the same exact moment, so it can only be a bad choice of stock footage by the show's makers.
Continuity mistake: Josh and another aide on the Mat Santos campaign are walking up the stairs to go out into a public area, as they reach the top of the stairs and about to go out the door they are not wearing sunglasses as obviously they are indoors, but in the next shot outside they are both wearing sunglasses without putting them on.
Other mistake: In the flashback episode where the early days of the first Bartlet campaign are shown, Toby is "teaching" C.J. the proper technique for passing a basketball, something she is not very good at. Yet in this episode, when C.J.'s sexuality is questioned in the news, her history as a star basketball player in high school and college is revealed.
Continuity mistake: When C.J. and Toby discuss and have a snack in Toby's office while the president waits for Chairman Farad's call, at some point C.J. talks while holding the pack of peanut butter crackers in one hand, and one that she is about to eat in the other. Less than a second and a half later, the camera is back on her, but she is now seen holding a bottle of water that was on the table only seconds before.
Continuity mistake: When Kate is in Leo's office with the red folder on her lap, the position of her hands changes depending on the camera angle. Viewed from in front both hands are gripping either end of the folder, from behind her hands are clasped together and rested on top of the folder.
Factual error: Near the end of episode 1 in series 6, General Alexander is briefing Leo, and the screen shows the Alpha strike headed east toward Syria from the USS Lincoln, which is depicted as being in the Mediterranean. In episode 2, he briefs the press, telling them that the strike was launched from the Lincoln in the Gulf.