To comfort someone is to give solace or to soothe. You might comfort your brother when his favorite team gets knocked out of the playoffs.
The verb comfort comes from the Latin word comfortare, which means “strengthen greatly.” To give comfort is to shore up the mood or physical state of someone else. It might take a long time to comfort your mother after her cat disappears. As a noun, comfort is anything that provides satisfaction or a relaxed and easy feeling. You might think that the comfort of your new shoes cannot be surpassed.