There are four main types of possibility. The distinctions become important in
philosophy, when things like an inverted earth, where the sky is green and grass is red, are one type of possible but not another.
Technological Possibility
A situation is technologically possible at time T if and only if the technology necessary to bring about the situation exists at time T. For example, on December 19, 2003, faster-than-light travel was not technologically possible, but launching artificial satellites into orbit was.
Physical possibility
A situation is physically possible iff it is consistent with the laws of physics. For example, near-lightspeed travel is physically possible, whereas two objects simultaneously occupying the same point in space is not.
Epistemic possibility
A situation is epistemically possible iff it is consistent with everything we know at time T. For example, on December 19, 2003, intelligent life on other planets was epistemically possible, whereas giraffes and elephants crossbreeding was not.
Metaphysical possibility
A situation is metaphysically possible iff it is consistent with the laws of logic (this type of possibility is therefore also known as logical possibility). For example, an inverted earth, identical to our own except that all the colors are inverted, is metaphysically possible, whereas an object that is both spherical and cubical is not.
All material taken from whatever I managed to absorb in my Philosophy of Mind class at the University of Rochester during Fall 2003.