A league has a specific meaning in the context of the big titles of professional Go, especially in Japan. It refers to the round-robin (all-play-all) used to determine the challenger who will play the incumbent (reigning) champion. Some leagues use other formats, such as the Swiss system.
Often, the league players are selected based upon two criteria: the top finishers of last year's league for the title and the winners of a set of single knockout tournaments. (Example: The Honinbo League consists of eight players: four are the top finishers of last year's Honinbo League; four are the winners of four distinct single knockout tournaments each open to one quarter of all professionals recognized by a Japanese certifying organization (Nihon Kiin or Kansai Kiin).
To win an all-play-all league, you must play well with either color, Black or White. In knockout tournaments, colors are decided randomly by nigiri each game and a lucky player may get his favorite color in nearly all games. Such a fluke can't occur in all-play-all format.
Many countries have team leagues in round robin format, such as China Weiqi League, Korean Baduk League and German Go Bundesliga. It's a convenient format for sponsors seeking some publicity because sponsoring a team is less costly and more flexible than sponsoring a whole tournament.