A public league is a league that anyone can join. The goal of this league is simple - draft a winning team!
A private league is a league that you can create that is not available to the public. For example, you could create a private league only for you and your friends. The goal is still the same - draft a winning team!
A theme league is a league with a twist. The goal is still to draft a winning team, but you can change the rules and regulations any way you want (provided it is feasible for the contest organizers to administer). For example, you could create a theme league where a rule is that teams can only draft players from 1980 or later. Or, you could have a combination of rules such as owners can only draft players from 1980 or later and the players must have played for Montreal. You can also limit options such as the Waiver Wire and minor league players. Once a theme league is full, the league founder has the ability to align the divisions anyway he likes. Theme leagues remain open until they are filled, but you may contact your theme league founder to see if he/she wants to merge the theme league with another league; or contact customer support to request that your team be moved to a public league or returned to the draft center so you can join a new league.