Weird Twitter is a loose and informal descriptor for a loose and informal group of Twitter accounts that post humorous tweets. It is very hard to describe, and I am doubtful that there is any actual definition that could reliably determine if any given account is "really" Weird Twitter.
"*record scratch* *freeze frame*
You're probably wondering how I ended up streaming video from a vinyl record" -- SQUARED COOL@qntm
Key components of weird tweets are that they are humorous, they make jokes using odd formats, often involve socially awkward or inappropriate situations, frequent puns, and/or surreal shower thoughts. But everyone has a slightly different list, many of which do not match mine much at all.
"Satan: And these are our pillars of tortured souls.
Me: Isn't that a bit much?
S: Well theyre load bearing pillars so its a necessary evil"
Weird Twitter is kind of almost sort of credited as an invention of Jon Hendren whose account, @fart, was and is a bit weird. He started his account in March 2008, but it wasn't until March 2012 that the term Weird Twitter was actually used by user @regisl (in the tweet "DESTROY NON-WEIRD TWITTER"). Since then, hundreds of accounts have drifted into the Weird Twitter aesthetic, whatever that might be.
"upon further inspection, the reports of mark twain's death are accurate."
Increasingly, the classic Weird Twitter accounts are becoming politicized, or mired in in-jokes, or simply become so much themselves that they don't quite fit the label of Weird Twitter. That's okay, though, there are always, always, always more where they came from.
"[first day as 911 operator]
ME: 911 what's your- your- *looks over to boss*
ME: hey, gotta go we have an emergency here"
-- rob elliott@rockymomax
Weird Twitter and Weird Sun Twitter are different things, and only peripherally related, at best.