What is Humanist Philosophy?

At its most basic, humanism involves any concern with humans (including human needs, human desires, and human experiences) first and foremost. Often, this also translates into giving human beings a special place in the universe on account of their abilities and faculties.
Working from this general conception, we can see that humanism is not a particular philosophical system or a set of doctrines, or even a specific system of beliefs. Instead, humanism is better described as an attitude or perspective on life and humanity which in turn serves to influence actual philosophies and systems of beliefs.
The difficulty inherent in defining humanism is summed up in the Encyclopedia of Social Sciences' entry on Humanism:

"Humanism as a technical term and as an intellectual or moral conception has always leaned heavily on its etymology. That which is characteristically human, not supernatural, that which belongs to man and not to external nature, that which raises man to his greatest height or gives him, as man, his greatest satisfaction, is apt to be called humanism."

"Humanism thus means many things. It may be the reasonable balance of life that the early humanists discovered in the Greeks; it may be merely the study of the humanities or polite letters; it may be the freedom from religiosity and the vivid interest in all sides of life of a Queen Elizabeth or a Benjamin Franklin; it may be the responsiveness to all human passions of a Shakespeare or a Goethe; or it may be a philosophy of which man is the center and sanction."

Humanism can also be better understood when considered in the context of the attitudes or perspectives it is normally contrasted against. On the one hand is supernaturalism, descriptive of any belief system which stresses the importance of a supernatural, transcendent domain separate from the natural world in which we life. Quite often this sort of philosophy describes the supernatural as being more "real" or at least more "important" than the natural, and hence as something we should strive for — even if it means denying our needs, values, and experiences in the here and now.
On the other hand are types of scientism which take the naturalistic methodology of science so far as to deny genuine importance or, at times, even reality to human feelings, experiences, and values. Humanism is not opposed to naturalistic explanations of life and the universe — on the contrary, humanists see it as the only viable means of developing knowledge of our world. What humanism does oppose are the dehumanizing and depersonalizing tendencies of modern science.
It is one thing to observe that humans are not valued by the universe at large, but quite another to conclude that therefore humans are not really valuable after all. It is one thing to observe that humans are but a tiny aspect of the universe and even of life on our own planet, but quite another to conclude that humans can have no important role to play in how nature progresses in the future.
We can conclude then that a philosophy, world view, or system of beliefs is "humanistic" whenever it shows a primary or overriding concern with the needs and abilities of human beings. Its morality is based upon human nature and human experience. It values human life and our ability to enjoy our lives so long as we don't harm others in the process.