a real world point of view
Assistant editors are responsible for maintaining a smooth flow of material from the production and post-production teams to the editor. Perhaps their most important duty is organising recorded material as it comes in so it can be accessed easily by the editor. In the course of their duties, they act as the editor's key liaison to other departments. They generally do not directly assist in the actual assembly of the rough or final cuts of episodes.
They are usually required from the first day of principal photography to the last day of post-production. Editors, by contrast, begin their work only after sufficient material has been filmed and organised, and may not fully begin work until after principal photography has wrapped.
On most projects, they are usually brought onto a project by the editor, rather than hired directly by the producer or director. However, BBC Wales seems to have a regular editorial staff who rotate duties on various episodes. It seems likely that various editing/assistant-editing teams are assigned to particular production blocks by the production manager. Due to the overwhelming workload on even a single production block, various editing teams are used throughout the course of a season.
This position has only been credited on-screen since the advent of the 2005 series of Doctor Who There may have been people who performed a similar function in the later years of the 1963 version of Doctor Who— indeed Tom Baker's wife, Sue Jerrard, was apparently an assistant editor on the original version of the programme — but the only editing credit given then was for film editors.
In other media
There are also "assistant editors" in audio and print. Big Finish Productions regularly credit assistant editors, who perform functions very similar to their television counterparts. They principally organise the various audio takes and cues so the editor can assemble them efficiently.
Assistant magazine editors similarly control the flow of articles and comic strips into the editor-in-chief's hands, who then decides how to compose the issue. However, assistant editors of magazines may also be responsible for verifying that articles have been written according to the magazine's manual of style, that the facts contained in articles are correct, or for excising material which runs overlong. As opposed to their television and audio counterparts, they may in fact engage in the actual work of editing, as the editor-in-chief wears several managerial "hats" not worn by his or her television counterparts.
Assistant editors for a publishing house such as BBC Books are the most distantly related to television assistant editors. They are typically engaged in giving a first read to a manuscript to help their editors understand whether it is worth their time to read the manuscript themselves, if the manuscript has not been commissioned. They may also provide primary editorial consultation to an author if the main editor is engaged in other, higher-priority work. Though similar to a television assistant editor as a conduit of information from the editor to other departments in the publishing house and to the actors who provide the content, they are much more engaged in the process of actually shaping the final product than a television assistant editor.