To flick dust off one's hat and congratulate each other
When you congratulate a friend on their promotion or new position, and they congratulate you back (because they'll probably use their connections to get you a job or a promotion as well)
Negative
他用不正当的手段坐上了公司的第一把交椅，他的同伙弹冠相庆 | Tā yòng bù zhèngdàng de shǒuduàn zuò shàng le gōngsī de dì yī bǎ jiāoyǐ，tā de tónghuǒ tánguānxiāngqìng
He used unscrupulous methods to rise to the head of the company, and his compatriots eagerly prepared for the promotions that would certainly be coming
[Source]
一帮弹冠相庆的小丑，不跟他们一般见识哦 | Yī bāng tánguānxiāngqìng de xiǎochǒu, bù gēn tāmen yībānjiànshì ó
That group of people uses their connections to gain power. Don't lower yourself to their level
[Source]
None
None
Chinese government officials used to wear distinctive hats, hence the emphasis on dusting off hats
《汉书》(Hàn Shū), a work documenting the history of the Western Han dynasty (西汉, Xī Hàn, 206 BC - 8 AD). It was written by 班固 (Bān Gù), and is generally known in English as the "Book of Han" or the "History of the Former Han."
王吉(Wáng Jí) and 贡禹 (Gòng Yǔ) were two very good friends from the town of 琅琊 (Láng Yá). One of the emperors of the Western Hàn Dynasty (西汉, 206 BC - 8 AD) appointed 王吉 to be a government official, but after some time he reversed his decision and dismissed him. 贡禹 suffered the same fate: he worked as an official in the imperial court, but after a few years, the emperor removed him from his post.
Eventually, a new emperor took the throne. This emperor apparently didn't have the same reservations as his predecessor, as he re-appointed 王吉 to his previous government position. It didn't take long for the news to spread to 贡禹, and when it did, he grew delighted — and not just for 王吉's sake. He quickly went and took his old government hat out of storage, dusted it off, and waited for his connection with 王吉 to turn into a job offer. Soon enough, the offer came through.