(written from a Production point of view)
Alien of the week is a notorious term often used by fans to describe a species which appears in an episode of Star Trek and may be featured prominently in that episode, but is never seen again. Of the Trek series produced so far, Voyager had the most, while Deep Space Nine, which was set aboard a space station rather than a starship, had the fewest. Some alien-of-the-week stories have spawned larger story arcs, most notably the Dosi in DS9's "Rules of Acquisition". However, the term is usually applied only to a single-appearance species when fans feel the episode did not live up to its potential or otherwise failed to meet expectations, and thus application of the phrase is open to interpretation.
A species from the Gamma Quadrant, the Dosi were contacted by Grand Nagus Zek, who sent Quark and Pel to open business negotiations in an attempt to establish contact with the Dominion. While the Dosi themselves were aliens of the week insofar as they were not featured in subsequent episodes, the Dominion and its war with the Federation would go on to be one of the most memorable story arcs in Trek history.
In its first season, some fans criticized DS9, the first series not set aboard a starship, for failing to make use of the Bajoran wormhole; rather than opening up new possibilities, the wormhole seemingly functioned as a device by which the writers brought aliens of the week to the station. Examples of such species include the Wadi, Hunters (along with the Tosk) and the Rakhari.
More often than not, "alien of the week" episodes are seen as fillers, or episodes written to fulfill the writers' quota for the season either due to lack of new ideas or in order to make time for the production of other, more involved episodes.
However, the term and its connotations are entirely subjective. The introduction of a new species, for example, would likely not be considered an alien of the week, and in some cases, an alien of the week will return in future or subsequent episodes. Examples of this phenomenon include the Borg, the Tholians and the Trill, all of which could be considered aliens of the week in their original context. Additionally, the term is not necessarily a pejorative one; the fact that a species does not appear in subsequent episodes does not necessarily diminish the quality of the story, although in some cases (such as the Voth), the phenomenon leads fans to believe a species' full potential was not realized.