Part of the larger movement in the humanities known at the nonhuman turn, posthumanism is an attitude or set of beliefs that positions itself against the human exceptionalism at the core of the Western worldview since the birth of Renaissance humanism. Generally speaking, there are two types of posthumanism: technological posthumanism, which celebrates what it sees as humans' inevitable transformation, via technology, into a new "posthuman" species, and ethical posthumanism, which argues for the legitimacy, dignity, and agency of the more-than-human world, animate and inanimate, earthly and cosmic. The kind of posthumanism we mean here is emphatically the latter.
While moving to challenge and replace the hubristic assumptions of humanism, posthumanism also objects to the social constructionism of deconstructive postmodernism, which again places the human at the center of all things. Additionally, in arguing for the agency of the nonhuman world, posthumanism represents a shift away from the epistemological obsessions of deconstructive postmodernism and toward an emphasis on ontology -- i.e. it's not so much how we know the world that interests us as what kind of world it is, and what is our true place in it.
But one hardly needs -isms and -ologies to understand the point. In short, posthumanism says: Enough about us. It's time for us to step aside and let the rest of the world speak. For with such a rich and abundant planet full of beings of all kinds and a whole cosmos pulsating with a vitality of its own, surely we have much to learn. A truly posthumanist future would be one in which our species participates in, but does not seek to dominate, the vast whole in which we are embedded.