I don't think that comparing the different types of IP and rights is that helpful -- actually I think it's more confusing than helpful, at least for purposes of US law.
Patent rights last for a relatively short period of time, but the protection that is given to patents is much, much stronger under the law. Even if someone comes up with the same idea completely independently -- too bad, it's patent infringement. There's no "fair use" exception to patent law. In return, the patent holder has to make the work public and after the patent term expires, anyone can use it. And patents are much harder to get as well.
By contrast, anyone can get a copyright on anything written down on a napkin or scratched into the side of a cave wall as long as its an "original work of authorship." (In fact, any such work is copyrighted as soon as it's set down -- you only need to register prior to filing a lawsuit.) Copyrights last for much longer but afford more limited protections -- specifically, only the enumerated rights in the statute. But copyrights, unlike patents, only protect the expression of an idea, not the idea itself. (Hence the reason you will be seeing ~500 thinly veiled ripoffs of Twilight in the next few years. God help us.) Copyright is also limited by fair use, the first sale doctrine, and other statutory constraints. It's also often harder to prove infringement.
What both have in common is their origin in the US constitution, which states that Congress may enact laws to protect intellectual property "To promote the Progress of Science and useful Arts, by securing for limited Times to Authors and Inventors the exclusive Right to their respective Writings and Discoveries." (Article I, sec. 8.)
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