to show off: to boast loudly and self-importantly, parading yourself for attention
Paul used this word exactly once, in the middle of his famous poem on love, and it appears nowhere else in the New Testament, nowhere in the Greek Old Testament, and nowhere in the Jewish writers of the period. It was not a common everyday word. It was a piece of moral vocabulary from the wider Greek and Roman world, and Paul reached for it on purpose to name a very particular fault: not just boasting, but the loud, showy self-advertising of the person who cannot stop drawing attention to themselves.
The key is that it describes a manner, not merely a claim. Ordinary boasting is about the content of what you say about yourself; this word is about the style: the swagger, the parade, the verbal peacocking before an audience. Ancient writers used it for a chatterbox envoy who bored everyone, for a student who preened over the famous philosophers he had read, and, in a private letter, for the great orator Cicero laughing at how he had performed and shown off in front of an important listener. The best short English words for it are brag, show off, or be a windbag.
Inside the love poem it sits right next to the phrase usually translated puffed up. The two go together as the outside and the inside of the same fault: showing off is the visible parade, and being puffed up is the inflated opinion of yourself underneath it. Paul aims the word squarely at the church in Corinth, which was preoccupied with polished speech and with putting its spiritual gifts on display. Love, he says, does not turn what it has into a personal advertising campaign.