to puff up: to swell with a pride that is all air and no substance
The picture behind this word is a blacksmith's bellows or a blown-up bladder: something filled with air so it looks big, while inside it is hollow. Paul takes that image and aims it at people. To be puffed up is to have an inflated opinion of yourself, your knowledge, or your spiritual standing, an opinion that looks impressive but has nothing solid underneath.
Paul almost owns this word. It barely appears in earlier Greek and is absent from the Greek Old Testament, yet he uses it seven times, six of them packed into a single letter to the Corinthians, where it names the one disease running through all their quarrels: a community swollen with self-importance. He then reuses it once in Colossians for a teacher who paraded humility while being secretly inflated by his own visions.
It is a single idea every time, so this entry treats it as one meaning. What changes is only the target of the deflation: partisan rivalry, a challenge to Paul's authority, smug tolerance of scandal, prized knowledge, or showy spiritual experience. In each case the force of the word is the same, the gap between how big a person looks and how little is actually there. That is why Paul sets it against two solid things: love, which is not puffed up (1 Corinthians 13:4), and building up, which raises a real structure instead of a swollen self (1 Corinthians 8:1).