Twitter is an American online news and social networking service on which users post and interact with messages known as "tweets".
Relevance to the Super Smash Bros. series
Nintendo uses this site to promote games and products. Nintendo also operates an account called Nintendo Versus, which focuses on competitive games, including the Super Smash Bros. games.
Many popular Smashers own Twitter accounts and often post various things about Super Smash Bros. on their accounts, including promotions for their Super Smash Bros. related YouTube videos, as well as comments about tournaments or even asking their followers questions or making a poll about the games.
Tournament directors also use Twitter to promote their event, tweeting out when and on what stream a tournament is starting, what bracket is being streamed, and sometimes results of a recent match, or the whole tournament.
From SmashWiki, the Super Smash Bros. wiki