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"
-- penjamin@upsidedowntrash
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."
-- brendle@brendlewhat
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*
BOSS: emergency
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.