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What does "invention" here mean? Is it limited to things that meet patentability criteria?I can't think of a case citation offhand, but my guess is that most (U.S.) judges would apply essentially the same principles to unpatentable "inventions" -- defined in 35 USC 101 as any "process, machine, manufacture, or composition of matter, ... [or] improvement thereof" [1] --- that qualified as confidential information of the employer.
(If an invention is unpatentable AND is public information, then normally it's fair game for anyone to use, at least in the U.S. --- EXCEPT that that (A) an employee has an implied duty not to compete with his employer while still employed there, and (B) the employee might be bound by a contractual covenant not to compete for a limited time after leaving employment, which might or might not be enforceable depending on the jurisdiction [California being a well-known case in point].)
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> is there a chart for other types of intellectual property like copyrights (most applicable to software, and hardware too)
I can't think of how copyright might apply to hardware, at least not to functional aspects of it. I suppose that aesthetic-design features of hardware might qualify as a "sculpture," which under 17 USC 102 (a)(5) would be eligible for copyright protection [2] But under 17 USC 102(b), that protection wouldn't cover functional aspects: "In no case does copyright protection for an original work of authorship extend to any idea, procedure, process, system, method of operation, concept, principle, or discovery, regardless of the form in which it is described, explained, illustrated, or embodied in such work." [2]
As to software, that depends on (A) whether the software counts as a "work made for hire," which in this context means that it was created by an employee within the scope of employment [3], and (B) if not, whether the employee signed an agreement giving the employer rights to the copyrighted work. [4]
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> ... trademarks, etc.
In the U.S., trademark rights arise from use of the mark in providing goods or services. (You can file an application to register a mark in the USPTO based on a bona fide intent to use the mark "in commerce that may lawfully be regulated by Congress," but the registration won't be issued unless and until the applicant shows that he / she /it has actually begun using the mark in Congress.)
That's not to say that an employer might not try to claim that a confidential idea for a particularly-good trademark belonged to the employer, e.g., because the employee who came up with the idea was working on a company project when she did so.
[1] http://www.law.cornell.edu/uscode/text/35/101
[2] http://www.law.cornell.edu/uscode/text/17/102
[3] http://www.copyright.gov/circs/circ09.pdf (scroll down to "Agency Law")
[4] http://www.law.cornell.edu/uscode/text/17/201