Wikipedia defines transhumanism as "an international intellectual and cultural movement that affirms the possibility and desirability of fundamentally transforming the human condition by developing and making widely available technologies to eliminate aging and to greatly enhance human intellectual, physical, and psychological capacities." That's true in a sort of reductionist sense, but I'm not sure it's a terribly useful definition.
If I were to define transhumanism, I'd say that it's an idea whose premise is that human nature is not some fixed quantity, forever unalterable; it's something that is a consequence of our biology and our environment, and it can be changed. Furthermore, advances in technology and in our understanding of biology, chemistry, and physics, give us the power to change it as we wish--to take evolution from a blind, undirected process to a process that we can make choices about. It's predicated on the idea that we can, if we so desire, choose what it means to be human.
A great deal of conventional thought has always held on to the idea that "human nature" is something that's a fundamental part of who we are, forever unalterable. Certain aspects of the human condition, from mortality to aggression, from disease to territoriality, have always been thought of as fixtures of the human condition; no matter how our society changes, no matter what we learn, these things have been assumed to be an immutable part of us.
Transhumanist thought holds that this isn't so. We are physical entities, whose nature comes from an extraordinarily complex dance of biochemical processes happening in our bodies. The way we respond to stress, the way we behave, the way our bodies suffer gradually increasing debility, all these things are the consequence of the physical processes happening inside our bodies and brains.
And they can change. Improved diet has made us qualitatively different from our neolithic ancestors--taller, longer-lived. Thousands of generations living in large numbers have made us more able to function in complex social environments; we have, in a sense, domesticated ourselves.
Right now, advances in biotechnology offer to revolutionize our view of who we are. What if aging and death were no longer inevitable? What if we could invent ways to repair genetic disorders? What if the human brain, which is a physical organ, could be modeled inside a computer? What if we could develop techniques to make our brains operate more efficiently? These sound like science fiction to a lot of people, but every single one of them is the subject of active research in labs around the world right now.
Transhumanism is a highly rationalist idea. It rejects the notion that human beings are corrupt, doomed to suffer and die as a result of a fall from grace. Rather, it postulates that the things that make us who we are are knowable and comprehensible; that the state of being human is a fit subject for scientific inquiry; and that as we learn more about ourselves, our ability to shape who we are increases.
The implications of these ideas are deeply profound. Transhumanist philosophy is built from the notion that things like indefinite lifespan, brain modeling, and improvement of human physical and intellectual capacity are both possible and desirable. Transhumanism, therefore, is profoundly optimistic.
It is not, however, Utopian. Like all new technologies, these things all have potential consequences whose outlines we can't see clearly yet. Therefore, transhumanism tends to be concerned not only with the possibility of biomedical technology but also its ethics; the study of transhumanism is, in large part, the study of bioethics. Who controls the direction of new, disruptive biomedical technology? What does it mean to be a "person;" is an artificial intelligence a person? How should new biomedical technology be introduced into society? How can it be made available democratically, to everyone who wishes it? What role is available to people who for whatever reason don't choose to benefit from new advances in medical understanding?
At its core, transhumanism is deeply pragmatic. Since it seems likely that biotechnology is going to improve over time whether we think about the implications of it or not, transhumanists think about things like bioethics, immortality, and the nature of consciousness in concrete, real-world terms, rather than as philosophical exercises. One of the things I most like about transhumanism is its drive to ask questions like "How can we maximize the benefit of what we are learning while maintaining human agency, dignity, and the right to choose?" Transhumanists are invited to be skeptical about everything, including the premises of transhumanism. It is quite likely that whatever views of the future we dream up will be flawed, as most prognostication tends to be. But by getting into the habit of examining these ideas now, and of considering the moral and ethical dimensions of our accelerating understanding of biology, we can at least train ourselves to get into the habit of asking the right questions as new breakthroughs come.