Copyright gives a creator control over a work of "art" - whether it's a short story, a musical performance, a movie, a piece of software or some other medium. Copyright is considered to exist from creation, though filing can make defending a copyright easier.

Trademarks give control over what we might call "unique identifiers" that contribute to the recognizable identity of a company or product - the term "Coca-Cola", whether or not in a particular swoopy font, the Harley-Davidson bar and shield device, and so on. Trademarks, I believe, have to be filed for to be protected under the law.

Patents protect inventions - the light bulb, the DVD, a "better mousetrap." Patents have to be filed for in order to be enforceable. A side effect of a patent is that the invention patented becomes known to the public; some might prefer to simply try to keep an invention secret in order to profit from it rather than reveal the details.

How do they relate? I'm not sure what that's asking. A patented invention can have a trademark-protected logo and be promoted by a copywritten piece of music, I suppose.... The three concepts cover different areas of what we now kind of lump together into "intellectual property."