A "tweet storm" refers to a series of consecutive tweets posted by someone, usually on a particular topic, in a rapid and continuous manner. It is essentially a succession of tweets forming a cohesive thread or discussion. Full definition
You see healthy scratches, guys getting pulled at random times, setting off panic tweet storms. (sbnation.com)
The new update brings a number of new tweaks, as well as a new feature called «Topics,» which makes it much easier to conduct live tweeting and create tweet storms. (macnn.com)
A few days before I sat down to write this piece, Joshua Lenon of Clio made the case much better than I could in a brief tweet storm on Twitter. (lawyerist.com)