In filmmaking terminology, a long take is one where the camera is continuously rolling through one shot, without any edits or cuts.
This sort of filmmaking has the advantage of time and efficiency, but also requires a lot of rehearsal and practice in order to get the take right. A mistake in the final thirty seconds of a five minute shot can be heartbreaking to a director.
The long take was perhaps most famously used in Alfred Hitchcock's tight drama Rope. In it, Hitchcock shot several takes exceeding 7 minutes, with some taking up the entire duration of the then-limiting factor of the ten minute reel. To mask the transition from reel to reel for these longer takes, Hitchcock would cut to a man's jacket or some other stationary image in order to mask the cut to the next reel. It is a very masterfully done movie, with lots of pans and zooms to capture the emotions of others.
Another movie that does the long take well is Woody Allen's Manhattan. In it, there is an extended sequence where four people sit at a table and talk. Rather than do quick cuts to focus on the speaker, Allen chose instead to suspend a camera above the conversation and rotate it almost constantly throughout the shot, stopping occasionally to focus on one or more of the actors. This created a much greater emotional connection in the film, and forced the actors to stay in character at all times, since they didn't know when they would be on-camera.
256 informs me that the first feature length movie to be shot in a single long take is Russian Ark, released in 2002. This sounds very promising! Also, television shows like ER make masterful use of the long take to switch between their varied storylines without cutting. Mike Figgis' movie Time Code, on the other hand, is a rather orgiastic display of long takes: the movie itself is split into four screens, each of them a 90 minute long take that occasionally cross paths with the other screens. It sounds a bit pretentious, but DejaMorgana assures me that it's "really impressive." So there you have it.
Today, the long take has gone the way of the buffalo, with the MTV-style of rapid-fire editing and the ever thinning attention span taking over at the box office. Today, many movies which prominently feature long takes (such as Ang Lee's The Ice Storm, John L. Sayles' Limbo, and the recent The Hours) are deemed "slow" and "dragging" by critics and audiences. It's a shame to see the eye of the camera downplayed as the stories and action in front take center stage.
UPDATE: Wow, a lot of you have mentioned some movies with excellent long takes in them, from the climactic fire of Tarkovsky's The Sacrifice to the snaking camera of Brian de Palma's Snake Eyes. I imagine there are a lot of movies with long takes worth mentioning, so please send them in! I'll make a list at the bottom.
Movies With Long Takes Worth Watching