The premise of the film is that Lincoln has an agenda item (the thirteenth amendment) he wants to push through Congress. He's recently been reelected -- after very publicly supporting emancipation -- and believes he has a mandate to see this agenda through. But he faces numerous obstacles. First, a Confederate peace envoy is offering to cease hostilities if slavery can be retained in some form; news of this will likely erode support for the proposed amendment. Second, his party, while maintaining large majorities in Congress, doesn't command two-thirds of the House, and members of the minority Democrats must be won over if the amendment is to pass. Third, his party is hardly united on the amendment; conservatives think it goes to far, radicals think it doesn't go far enough, and none of them like him forcing this on a lame duck Congress. Fourth, Lincoln's own views on slavery and the war have evolved over his first term, and many in Congress and in his own cabinet distrust him as a result.
These struggles are the essence of the American presidency. And the film nicely portrays both the powers and the limitations of the president. It makes the point that should be so obvious but is so rarely portrayed in political films: the president has no direct power over Congress. He is not a member of it, he cannot author bills, he cannot force Congress to consider a bill, and he cannot (despite what the creators of "The Contender" would have you think) demand a roll call vote. The president runs and is elected on an agenda but is largely dependent on Congress to see it through. The film also notes that the president can't dictate to his party: Preston Blair, one of the founders of the Republican Party, makes far more demands on Lincoln than the other way around, and Lincoln basically begs Thaddeus Stevens and the Radicals for their support. And in terms of the president's legal powers, Lincoln himself is shown wrestling with whether his Emancipation Proclamation was actually constitutional or whether it would have any authority in peacetime. He well knew that he was exploring uncharted and potentially dangerous areas of the law and was unclear about his power to do so.
But the president does have other powers, notably the power to make patronage appointments and control the military. He can influence media coverage but can't control it. And while we do see a few examples of the president attempting to personally persuade some members of Congress, it's not clear how effective that is, and this isn't remotely treated as his most important power. (A lesser film would likely have shown the president using his bully pulpit powers, but that would have been both ahistorical and stupid here.)
I'm open to suggestions here, but I have a hard time coming up with another film about the presidency that gets at these core issues of executive limitations and powers. "The Contender" was a joke in this regard. "All the President's Men" is great but is basically about the media. "The American President" is pretty much a romantic comedy. It does show the president struggling with pushing bills through Congress, but largely resorts to magical bully pulpit powers in the end. "Dave" is lighthearted comedy. "Seven Days in May" addresses some of these issues but almost completely ignores Congress. The one film that handles these issues seriously, I think, is "Advise and Consent," which chronicles a president's difficult nomination of a new secretary of state, although much of that film's focus is on the blackmailing of a particular senator rather than on the president, who disappears for much of the film. "West Wing" actually addresses a number of these issues in a serious way, although scattered across many different television episodes.
So I plan to use "Lincoln" in my film class, and I'm grateful for a film that finally deals with the executive branch in all its glory and shortcomings.
Update: Here's the actual Thirteenth Amendment, featuring an unusual and unnecessary presidential signature. (via William Adler on Twitter)