What it means
To drop a clanger is to make a big, obvious blunder, usually by saying exactly the wrong thing out loud and letting it crash round the room. It’s very British stuff: bad timing, wrong name, daft comment, then that grim little silence where everyone feels the social bell still ringing.
Usage examples
"Down the pub I asked my mate’s missus when the baby was due. She held up a pepperoni pizza and I knew I’d dropped a clanger."
"I really dropped a clanger at the party, asked when the baby was due and she was not even pregnant."
"I dropped a right clanger in the work meeting and congratulated Dan on the promotion he'd literally just been told he didn't get."
"At Sunday lunch my uncle called his new girlfriend by the ex-wife's name and, mate, the bloke absolutely dropped a clanger."
"I dropped an absolute clanger in the group chat and sent the moany message about my manager straight to my manager."
Where it comes from
It grows out of clanger, a word for a loud clang or resounding noise. By the late 19th century in British English, drop a clanger was being used for a glaring mistake or verbal blunder. The picture’s the whole joke: one stupid slip, massive noise, everybody notices.