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.
Starting around the launch of Ultimate, high level players began to notice the prevalence of misleading clips of combos performed by various fighters, which gave the impression that the fighters in question had a significantly stronger combo game that what could practically be achieved in competitive play. They were usually performed against low level CPUs in Training mode, but were sometimes taken from online play as well, and a new wave of them would often be posted every time a new fighter was released. These were given the sarcastic name of "Twitter combos", and are often regarded as dubious by high level players.