Members: someone who belongs to a mailing list run by your Lyris Server. A mailing list member has no Lyris administrative powers.
List Administrator: typically, someone (or several people) on each mailing list act as the administrator of the mailing list. This means that they can add and remove members of the group, approve moderated messages, and other things in his vein.
Site Administrator: a site administrator typically doesn't deal with the day-to-day workings of a mailing list. Instead, the site administrator can create and remove mailing lists, create documents, create auto-responders and other functions which affect the site as a whole. A Site Administrator is also entitled to act as the list administrator of any mailing list in their site.
Server Administrator: the server administrator can create sites, and has control over things which affect all the sites on their server. A Server Administrator can access all Site and List administration menus.