Someone who's dogged is stubbornly persistent. In the old Looney Tunes cartoons, Wile E. Coyote's pursuit of the Road Runner is dogged. He simply will not give up.
You can see the word dog within the word dogged. It's no trick. If you're dogged, you are as obstinate and tenacious as a dog who smells a bone. An earlier definition of this adjective was more general, meaning "having the qualities of a dog." Today, though, if someone describes you as dogged, they simply mean that you won't stop until you get what you want.