So, we have:
*Logical possibility is usually considered the broadest sort of possibility; a proposition is said to be logically possible if there is no logical contradiction involved in its being true. "Dick Cheney is a bachelor" is logically possible, though in fact false.
* Metaphysical possibility is either equivalent to logical possibility or narrower than it. For instance, "Water is H2O" is metaphysically necessary but not logically necessary.
* Nomological possibility is a possibility under the actual laws of nature. Most philosophers since David Hume have held that the laws of nature are metaphysically contingent -i.e., that there could have been different natural laws than the ones that actually obtain. If so, then it would not be logically or metaphysically impossible, for example, for you to travel to Alpha Centauri in one day; it would just have to be the case that you could travel faster than the speed of light. But of course there is an important sense in which this is not possible; given that the laws of nature are what they are, there is no way that you could do it.
* Temporal possibility is a possibility given the actual history of the world. Barack Obama could have chosen a degree in Accounting rather than Law, but there is an important sense in which he cannot do it now. So, the "could have" expresses the fact that there is no logical, metaphysical impossibility involved in Obama's having a degree in Accounting instead of Law; the "cannot now" expresses the fact that that possibility is no longer open, given that the past is as it actually is.
