During the cold open President Bartlet delivers a speech in which he references a civil procedure professor he had. Civil procedure is a first year course at most law schools. Bartlet received a B.A. in American studies and a minor in theology from Notre Dame and a Masters and a Ph.D. From the London School of Economics. He never attended law school and would never have had reason to have a civil procedure professor. This may be explained as rhetorical flourish but it is most likely that the biography of the President not being fully fleshed out at this point in the first season.
Leo's house is seen when his car pulls up to it and it is clearly a different house than we see Leo living in in The Pilot.
While the staff is eating Chinese food, there is a plate of food on the table that we assume is either Josh's or Leo's. When Leo comes toward the table and the shot changes, suddenly the plate is covered with foil. Then, just as suddenly, Sam is eating from the plate.
While talking about financial disclosure near the beginning of the show, Toby makes the remark that women can't get enough of his 1993 Dodge Dart. Dodge stopped making the Dart in 1976, and didn't resurrect the model again until 2013.
During the show, Josh refers to a "Rutger Mini 14" assault rifle. The rifle is in fact a "Ruger Mini-14". Later, an off screen reporter also mentions a "Rutger 14" Additionally, the written script refers to a "Tech 9" vs. the real weapon "Tec 9". The latter difference obviously doesn't manifest in the spoken version.
The water brought to Leo, while he's conferring with the vice president, has ice that doesn't float.
When the gang leaves the venue after the president's speech at the beginning of the show, they take a basement route, encountering several employees including a couple making out. Secret Service would surely have cleared the stairwells and corridors prior to leading the president through them.
The steadicam and its operator David Chameides can be seen reflected in a pane of glass as Chameides passes Allison Janney and Bradley Whitford during the opening walk-and-talk scene.
The newscaster voice-over pronounces the firearm manufacturers name of Ruger (Roo-ger) as Rugger.