Introduction to Philosophy/Philosophy of Science
There are two different meanings of the term philosophy of science: the first one is about the scientific method and it is the one meant in this article; the second one is about some philosophical questions that are answered by scientific discoveries. In recent decades physics have changed our concept of reality and biology changed our concept of life, philosophy of science in this meaning is treated in A scientific perspective of philosophy.
There are subdisciplines of philosophy of science which take a philosophical look at issues arising from specific scientific disciplines: for instance, philosophy of biology looks at the meaning of concepts used in evolutionary biology, while the philosophy of social sciences looks at topics like what a 'theory' is in a field like economics or sociology.
Karl Popper 
The founding father of this field is widely considered to be Karl Popper, writing at the beginning of the 20th century. The idea that one was necessary stemmed from observation of the Freudian claims on the Oedipus complex, essentially Freud's theory cannot be proven wrong by any means. Whilst this might sound like a positive aspect of the theory, it means that there are also no tests to verify the claims the theory makes.
Popper's proposed means of putting this right was to apply the principle of falsificationism to any one theory. To be scientific the scientist in question has to try to prove his theory wrong, or at the very least put forwards tests that can be shown to be either correct or incorrect. This led to the problem of when a theory can be known to be truly scientific and as such a correct interpretation of the natural world. Popper's answer to the criticism was that each test corroborated the theory further. This to many is simply the equivalent of the inductive "All swans are white" argument, i.e. the observation is correct until you happen to pass by Australia and find a black swan.
Later theories include Kuhn's theory of a scientific revolution. This proposes the theory that scientists will work within one given model, or paradigm, of the universe and make this useful and predictive until such a moment in time when a new idea or theory overthrows the old system, such as Galileo's observations of the solar system.
Feyerabend however rejected the idea that scientists are rational creatures entirely in his book "Against Method" where he argues that scientists do not in fact follow any one given method but are much less rational than their propaganda makes them out to be.
Thomas Aquinas 
The ideas put forth by Popper, Kuhn, and Feyearbend, wrestle with an issue mentioned by Thomas Aquinas. He said, "...other sciences do not argue to prove their own principles, but argue from their principles to prove other things which the sciences include..." (Summa Theologica article eight or ). While he wasn't directly talking about the philosophy of science, his words are instructive of the present topic. If science cannot argue to prove its own principles, then something else must or science has no basis.
Imre Lakatos 
A fourth contemporay is Imre Lakatos, what he gained most renown for in the circles of philosophy of science was his attack on what he terms as pseudo-science. His demarcation lies in whether any given program is progressing or degenerating. As such the program that is producing most new knowledge is the one that will be favoured by scientists as the one closest to the truth.