Something that's mighty is large and powerful. Mighty Mouse is no ordinary mouse; he's a superhero. It's best not to mess with anyone with mighty in their name.
A mighty beast in a fairy tale is ferocious and intimidating, and a mighty power in the business world exerts great influence. Colloquially, you can also use mighty to emphasize something, or to mean "extremely." This informal use is especially common in the U.S. South, where a host might say, "We're mighty glad to have you!" Mighty comes from the Old English root mæht, "strength or power."