What is philosophy?
A good way to get a grip on this question is to look at an example. Consider this passage from Plato's Meno:
Meno: [...] Is this true about yourself, Socrates, that you don't even know what virtue is? Is this the report that we are to take home about you?
Socrates: Not only that; you may say also that, to the best of my belief, I have never yet met anyone who did know.
Meno tries to refute Socrates by providing an explanation:
Meno: But there is no difficulty about it. First of all, if it is manly virtue you are after, it is easy to see that the virtue of a man consists in managing the city's affairs capably, and so that he will help his friends and injure his foes while taking care to come to no harm himself. Or if you want a woman's virtue, that is easily described. She must be a good housewife, careful with her stores and obedient to her husband. Then there is another virtue for a child , male or female, and another for an old man, free or slave as you like; and a great many more kinds of virtue, so that no one need to be at a loss to say what it is. [...]
Socrates: I seem to be in luck. I wanted one virtue and I find that you have a whole swarm of virtues to offer. But seriously, to carry on this metaphor of the swarm, suppose I asked you what a bee is, what is its essential nature, and you replied bees were of many kinds, what would you say if I went on to ask: 'And is it in being bees that they are many and various and different from one another?'
This piece of dialogue is a paradigm example of philosophising. Why? Socrates questions whether we really understand things which we take for granted in common life or in scientific thought. (We all know what virtue is! After all it is pretty clear who is virtuous and who is not.) He raises questions in order to achieve a deeper and more general understanding of basic but unexamined notions like virtue, justice and knowledge. This is one of the main aims of philosophy. Socrates pursues this aim in a characteristic way: he rejects answers given to him by constructing arguments, not by conducting experiments; he proposes definitions and checks them against counter-examples. Crucial for philosophical progress is the engagement with reasons in dialogue.
The Department's approach to teaching follows the dialectial nature of philosophy just exemplified. Philosophy cannot be taught without dialogue. Participation in philosophical analysis and argument are essential for learning philosophy. When you join the Department, you will explore questions like "What is knowledge?" or "Does the value of an action consist in its consequences?" in small groups and tutorials.
If you would like to get some idea of what philosophy is like, you might look at some of these books.
Simon Blackburn, Think, Oxford 1999. Also: Being Good: A Short Introduction to Ethics, Oxford 2003.
A.C. Grayling (editor), Philosophy 1: A Guide Through the Subject, and Philosophy 2: Further Through the Subject, Oxford 1998. (A topic-by-topic introduction to philosophy.)
John Perry, Michael Bratman and John Martin Fischer (editors), Introduction to Philosophy: Classical and Contemporary Readings, Oxford 2006. (Anthology with classic and recent papers.)
Jay F. Rosenberg, The Practice of Philosophy: A Handbook for Beginners, Upper Saddle River, NJ, 1995. (Gives you an idea how to do philosophy.)
Bertrand Russell, The Problems of Philosophy, Oxford 1997. (Introduces the reader to central epistemological and metaphysical problems and offers controversial solutions.)
P. F. Strawson, Analysis and Metaphysics: An Introduction to Philosophy, Oxford 1992. (A non-historical introduction focused on central philosophical concepts.)
George Berkeley, Three Dialogues between Hylas and Philonous. (A dialogue which takes you through the basic arguments for and against idealism.)
René Descartes, Meditations. (A central and very accessible text.)
Immanuel Kant, Prolegomena to any Future Metaphysics.
Plato, Republic. (The best introduction to Plato is Plato.)
John Stuart Mill, On Liberty.