An underwriter is the person — or the company — who decides whether to take on someone else's risk, and on what terms. In insurance, which is the main use, the underwriter is the one who looks at your case (your age, your health, your car, your factory, your ship) and decides whether the company will insure you, how much you must pay, and what will not be covered. So when your insurance application is "with the underwriters", it is being judged, not just processed. The same word is used in finance: when a company sells shares to the public for the first time, an investment bank acts as underwriter — it promises to buy whatever shares the public does not, so the company is sure to get its money. In both cases the underwriter is the one who says "if it goes wrong, that lands on me", and charges for standing there. This is a workplace word, not a conversation word: you will meet it in contracts, job titles, and business news far more than in speech. Though actually, in Taiwan the everyday face of insurance is the salesperson who visits you; the underwriter is the invisible person behind them who approves or refuses the policy, which is why students often have not met the idea even in Chinese.