a real world point of view
A runner — or in some BBC Wales credits, an un-numbered assistant director — is an entry-level job in the production of shows set in the Doctor Who universe. As the name suggests, they run errands —generally those that no one else on the production has time to do. Their precise job is ill-defined, as their role changes depending upon the needs of the day. Nevertheless, they are distinct from the related floor runners. Both of these types of runner are similar to, but distinct from, the production runner, who is sometimes, in at least modern British parlance, called the production assistant. All breeds of runner exchange low wages and long, irregular hours for invaluable work experience and jobs that are never the same from one day to the next.
A simple "runner" tends to be mostly focused on jobs that will take him or her away from the production office or the studio floor. Most of the errands the runner is responsible for take them away from the actual studio or location itself. If equipment needs to be hired from an outside source, for instance, the runner will likely be involved in the physical transport of that equipment.
Floor runner Edit
In contrast, floor runners leave the studio less often. They may be tasked with things like bringing tea and coffee to the on-set personnel or helping to transport material within the studio complex itself. Floor runners generally report to an assistant director. A common career path is for floor runners to ascend to the position of director, though this hasn't happened yet on the relatively young production team at BBC Wales. Nevertheless, career development towards this goal can be detected at BBC Wales. Heddi Joy Taylor, for instance, was a floor runner during series 3 and series 4. She rose to the position of 3rd assistant director on The Next Doctor and has continued that role during Matt Smith's first series.
Production runner / production assistant Edit
A production runner, though a similar sort of "gofer" to other runners, is allocated to the production office itself. Like the runner, he or she will infrequently work on the studio floor. Their job may require such things as filing paperwork, answering telephones, arranging lunches, making sure that transportation details are implemented, and distributing production paperwork. They often work closely with the production secretary in these tasks. They do sometimes work on-set as well, though perhaps more often on location. In this capacity, they often liaise with the 3rd assistant director in the management of supporting artists (or extras). They can also be employed to provide some minimal crowd control, in conjunction with civil authorities, and in cleaning up locations after a shoot has been completed.
A production runner may report to the producer directly, or to another senior member of the production staff, like the production co-ordinator. Production runners often advance through the production ranks, or sometimes use the position as a springboard into other departments.  Tim Hodges, for instance, began on Doctor Who as a production runner in series 1, but by the time of Blink had become an assistant editor. Anna Evans also started out as a production runner in series 1, but was 2nd assistant director on Blink, and the regular 2nd AD on series 1 of The Sarah Jane Adventures.
The BBC Wales version of Doctor Who has not typically credited both production runners and production assistants on the same episodes. However, both positions have been credited by BBC Wales. It is somewhat unclear whether BBC Wales see any meaningful distinction between the two jobs. Certainly most job descriptions don't reveal any substantive difference.
The 1963 version of Doctor Who Edit
Someone credited as a production assistant during the original run of Doctor Who was not a runner of any kind, but someone much farther up the production hierarchy. According to Philip Hinchcliffe and Marion McDougall, a PA during their era was what the BBC Wales show calls a "first assistant director". (DCOM: The Android Invasion) For this reason, production assistants from the 1963 version of the programme are listed separately on this wiki.