1.
official business, errand, order
Business or errands, especially those conducted on behalf of the government, the imperial household, or a person of high status. The 御 prefix adds formality and respect.
2.
arrest, capture
The apprehension of a criminal. Historically, the cry 「御用だ!」 was used by Edo-period constables when making an arrest, and it remains well known from period dramas.
Usage
Sense 1 uses 御用 as a polite term for business or errands. The compound 御用達 means "purveyor to" (historically to the imperial household, now used more broadly for high-status endorsement). 御用聞き originally meant a servant who takes orders, but now can refer to a delivery person from a local shop.
Sense 2, 御用になる or 御用だ, is a classic expression meaning "to be arrested." Iconic from 時代劇 (period dramas).