What it means
Unflappable is for someone who stays cool and steady when everything around them's going sideways. Fires to put out, people flapping, stress in the air, and they're still talking in a normal voice and sorting it. It's proper praise for the person who doesn't get rattled, doesn't panic, and keeps the whole circus from turning into soup.
Usage examples
"Nothing fazes her, she is completely unflappable in an emergency."
"Our captain is utterly unflappable, even two goals down he never panics."
"The pilot stayed completely unflappable through the storm, calm voice on the tannoy like he was ordering a coffee."
"You want an unflappable mate in a crisis, someone who just makes tea while everyone else loses the plot."
"Everyone else was stress-texting in the group chat, but Priya was totally unflappable and had the whole mess sorted by lunch."
Where it comes from
It builds off flap in the British sense of panic or agitation, a use that was already knocking about by the early 1900s. Stick un- on the front and unflappable lands as the opposite, for someone who doesn't get thrown or worked up when everyone else is having a proper wobble.