Cutting dies let you cut clean, repeatable shapes that would be impossible by hand: frames, tags, silhouettes, decorative borders and small pieces with fine detail.
They're used with a die-cutting machine and work on paper and cardstock as well as thicker materials, depending on the die. They're an investment that pays off for years, because a good die will end up in dozens of different projects.
If you're starting out, frame and tag dies are the ones that get used the most.