Slugs are a way of identifying some data within Ditto in a form that is more "human friendly". For example, a list may have an internal id of 327, but this is more understandable as its slug counterpart of Movies-with-the-Best-Movie-Posters. Any aspect of Ditto that needs to be expressed in a URL will have a slug, including lists, people and entities.
One thing that you will need to be aware of is that slugs can change. Although the most likely form of slug to change are user slugs (whenever a user changes their name, their slug will change accordingly), any slug may periodically be modified.
As a consequence of this, you should not cache or rely on a slug staying constant. Instead you should only persist the internal ID of the data you are interested in, e.g. the ID of the list or user. You can subsequently use the Person.GetById or List.GetListSummaryById methods to retrieve the current information for whatever you are interested in. You can then use the retrieved slug to request any further information.