A sub-series is a series that is entirely included within a parent series. An example is The Sims series, as every entry in the series is also included in the general Sim series. For a list of sub-series, see: Category:Sub-series.
Sometimes sub-series are labeled as spin-offs, but as far as classification on this Wiki goes, a spin-off is a series that's not included within its original parent series. One example is the Chocobo series, a spin-off of the Final Fantasy series. It is not included in the Final Fantasy series page, as it has elements that mostly originate from its own line of games that aren't featured in Final Fantasy games, so it's given its own page.
If a sub-series gets enough entries or is very significant, it might get its own article that's divided from the parent series, featuring its own links to other series, and also including links to other sub-series of the parent series.
It is also possible that a sub-series can become a spin-off series after a while. This happens when a sub-series gets one or more releases that can't be included in the parent series. One example of this is the Rabbids series of games; originally created as a sub-series of the Rayman series with the release of Rayman Raving Rabbids and Rayman Raving Rabbids 2, after the release of Rabbids Go Home onwards, the series stopped featuring elements from the Rayman series, so it started being considered a spin-off instead of a sub-series from then on.