Trademarks are distinctive signs or symbols used by a person or company to promote and differentiate their products and services from the competitors. They typically consist of a name, word, phrase, logo, or combination of the four, and are distinctly associated with their brand. See the article on the subject for an in-depth explanation.
Trademarks belong to a category called "intellectual property rights", alongside similar concepts called copyrights and patents. A full discussion of their definitions and purpose is beyond the scope of this entry (we have a separate comprehensive article explaining how trademarks works, click here for details), but the three terms all bear a common theme: They acknowledge creation or ownership of something and provide the owner with some control over how it gets used. If somebody else attempts to use it commercially without the owner's permission (often with an exchange of money involved for such permission), the owner can take them to court and sue for damages.
Now, in Real Life, there are certain rules and limitations on what does and does not constitute an infringement, and what is and is not eligible for protection under intellectual property rights laws in the first place. These laws are also what (ostensibly) protect independent artists, inventors, and writers from having their work stolen or destroyed by Evil, Inc., Executive Meddling, severely Misaimed Fandom, or having the Serial Numbers Filed Off.
But there are no such limitations or ethical quandaries in fiction! Some corporations (for example, MegaCorp) are so damn powerful and wield so much influence that they can put a stamp of ownership on literally anything. The grass in your front yard? Patented by a gardening company. The morning sunrise? Copyrighted. The name of your favorite pet? A bright sunny day? The word "the"? Yes, them too. If it exists, then somebody, somewhere, has stamped a copyright or trademark upon it, regardless of common sense or reason, and they'll happily send out their Army of Lawyers to collect royalties at even the slightest hint of infringement.
It should be noted that not all copyright and trademark claims are valid; just look at the mess over "Happy Birthday to You!" as an example of a company attempting this and (eventually) failing. Often leads to getting Screwed by the Lawyers.
Related to Stuck on Band-Aid Brand, where a company not only holds the trademark but continually remind people of it. May overlap with Trade Snark, where words are labelled as trademarks for humorous purposes.
Don't confuse this for tropes that the Walt Disney Company owns at using, such as the Disney Death or Disney Acid Sequence. It's more general and metaphorical than that.
