What it means
The supposedly dominant, confident, take-charge type who leads the pack, mostly used about men and increasingly with an eye-roll. Going full alpha is striding in and bossing the room. The whole idea leans on a wolf-pack theory that turned out to be wrong, so it now carries a heavy dose of irony.
Usage examples
"He stood at the front barking orders, going full alpha at a kids' football match."
"Booking the restaurant without asking, very alpha of you."
"He's been in the office six minutes and already doing the alpha routine, chin up, voice loud, acting like he owns the kettle."
"Some bloke called himself an alpha on a first date and the whole table spiritually evacuated."
Where it comes from
It comes from alpha in the Greek alphabet, which got borrowed into old animal hierarchy talk for the supposed top male in a pack. That wolf model was later shown to be shaky, but alpha survived online and spread hard through pickup, self-help, gym-bro, and manosphere chat, now often with a big wink.