SCL (The Society of Composers and Lyricists) is the oldest and most established organization for composers.
The Society of Composers and Lyricists (SCL)
ASMAC is another excellent organization:
American Society of Music Arrangers and Composers
GANG (Game Audio Network Guild) is a very active organization focused on game composers.
Game Audio Network Guild (GANG)
ScoreCast is a fairly new and very active organization with chapters in various countries:
Film Music Foundation has outstanding video interviews and other resources:
Film Music Foundation
Film Music Network is mostly inactive now, but they do have some nice resources:
Film Music Network
I don’t know anything about The Academy of Scoring Arts, but it looks cool:
The Academy of Scoring Arts