Continuity mistake: Season 7, episode 2, "The Mommy Problem." At the end of the episode, when Matt Santos is approaching his fighter jet, the number "15" is on the left part of the jet's nose. Later, when the jet is taxiing towards take-off, the number changes to "01".
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.
Continuity mistake: About 18:59. Right after Josh hugs Annabeth, she gives him a cell phone to talk to the White House. In the shot from behind, you can see that he is holding the phone backwards with the screen facing out. When they go back to this shot (after a shot from the front), he is holding it properly.
Continuity mistake: After Louise Thornton leaves the Democratic media consultant meeting, Josh runs after her inside the Santos campaign HQ. At some point he talks to her while catching up on her and ends the shot walking ahead of her. In the very next shot, he is again behind her, walking at a fast pace to catch up.
Continuity mistake: Josh comes back from his trip to visit Sam. He walks in the door and lets his bags drop on the floor. He collapses on the couch with his arms spread wide. Immediately, there's a knock at the door and he opens his eyes. When the angle changes towards the door showing him going to get up, his arms are now crossed above and behind his head resting on the couch arm.