SESAC is the only PRO in the U.S. that is not open to all songwriters; instead, you must receive an invitation to join. SESAC represents over 400,000 songs on behalf of its 30,000 affiliated writers. “With an international reach and a vast repertory that spans virtually every genre of music, SESAC is the fastest growing and most technologically adept of the nation’s performing rights companies,” reads the SESAC site. Note: the SESAC abbreviation is, today, meaningless; the organization was originally founded to serve European composers underrepresented in America before branching out to become a full service PRO.
Music Publishing is the business of protecting and promoting song copyrights and collecting the royalties that these copyrights generate. Think of a song as intellectual property (it's an original piece of art!), and music publishing as the business that makes sure that songwriters get paid a fair compensation when their intellectual property is used by companies. Those companies could be record labels (mechanical royalties), radio stations, bars, and restaurants (performance royalties), or film studios and advertising agencies (sync license fees), and many more.