A vat was a large tub used for holding liquids.
Leonard McCoy's analysis, in 2266, of Schmitter's remains of largely bone and teeth, the doctor compared to the conditions those fragments might have experienced to as if he had been thrown into a vat of extremely corrosive acid. (TOS: "The Devil in the Dark")
In 2370, Grand Nagus Zek sent Quark on a mission into the Gamma Quadrant to purchase 10,000 vats of tulaberry wine from the Dosi, however, the Dosi could only offer 5,000 vats. Later, when it appeared negotiations were about to be completed, the Nagus changed his purchase offer from 10,000 vats, which Quark wasn't sure they were even going to agree to, up to 100,000 vats. The larger offer was rejected by the Dosi, who later explained that there was not that many vats on the entire planet. It was then explained by Zyree that the only people who had access to that much tulaberry wine were the Karemma. (DS9: "Rules of Acquisition")
Kor believed that the modern Emperor Kahless to be "a pretender, a clone of the original Kahless cooked up in a vat by ambitious clerics. And if you hadn't supported him, he'd be right back in that vat where he came from." (DS9: "The Sword of Kahless")