Other forms: trademarks; trademarked; trademarking
A trademark is a distinctive feature of something or someone, and to trademark is to brand something, especially in a legal sense.
You know how a comedian often has a particular joke she's famous for? That's her trademark. A trademark is something specific to a person that can be used to identify her. The legal sense of trademark is an official representation of a business or product that only they can legally use — like the Nike swoosh or their motto "Just do it."