When something is pursuing another thing, it is following or chasing it with the intent to catch. Cops are often pursuing criminals.
You may be literally pursuing your three-year-old niece in a game of chase or pursuing a friendship with the neighbors who own a trampoline. In movies and TV shows, the good guys will be shown pursuing the bad guys. At the park, you can see dogs pursuing squirrels. Anyone pursuing something wants that thing badly. On the way to an interview for a job you're pursing, you might have to start pursuing your hat if it flies off your head.
