a real world point of view
A 1st assistant director — variously known as the 1st AD and the first — is the director's chief assistant. He or she is the supervisor of the rest of the directorial staff, and therefore is responsible for the work of the 2nd AD, 3rd AD and most runners. During the 1963 version of Doctor Who, this position was usually credited as a production assistant, a term that, confusingly, is now equivalent to a runner. (DCOM: The Android Invasion)
Beyond this supervisory role, though, the first is specifically responsible for the assembly of the shooting schedule. (CON: "A Brush with Genius") This is usually accomplished by consulting with the director, storyboard artist, and, if necessary, pre-visualisation technician to compile a shot-by-shot plan of the entire episode. They then must estimate how much time it will take to achieve each shot and create a timetable for completing the entire episode. Once the schedule is devised in pre-production, it is up to the first to ensure that principal photography proceeds according to that plan, ever mindful that the schedule has budgetary implications. In the same way that the production manager turns the producer's plans into concrete action, so, too, does the first take care of the practical details of filming so that the director's creative choices can be brought to life.
One of the most obvious ways in which the first controls the pace of filming is by "calling the roll". It is usually the first's voice that is heard actually directing the camera department and actors to proceed through filming. The first will call for "quiet on the set", or some variation thereof, to indicate that filming is imminent; instruct the camera and sound departments to "turnover", or begin rolling the film and audio stock; and, perhaps more famously, to call "action" to get the actors to begin the scene. Of course, the director is ultimately in control of photography, and may choose to make these calls personally, especially that of "action". Graeme Harper, in particular, seems to prefer to be the one to call "action", judging by outtakes from Doctor Who Confidential.
The first (sometimes through the second or third) is also the usual point of contact between the director and the rest of the production team. He or she will make sure (usually through the second) that the production office has daily reports of the progress of filming. Actors and other production personnel will usually be kept ready to make their contributions to a shot by the first and his or her assistants.