What is philosophy?
Very crudely, we might say that the subject matter of philosophy is the nature of reality at the most abstract level. Slightly more specifically, the central topics philosophers are interested in are these: truth, thought, language, knowledge, and value. Philosophers study these topics, and try to find theories that make good sense of them.
How does this relate to science?
In some respects, philosophy is like science. Like scientists, we try to offer theories that make sense of the world. But the difference lies in the methodology. Scientists appeal not only to logic and reason, but also to sensory evidence. Even if philosophers do these same things, we do them to different degrees: we appeal more to logic and reason than we do the senses.
So philosophy is similar to maths?
In methodology, yes. Both subjects investigate the world using logic. The main differences between the two are their subject matter, though even here there is sometimes overlap.
I thought all philosophers died long ago? Don't you just study what they said?
No, and no. There are plenty of philosophers around today, and we study the very same issues that the great dead philosophers investigated. Of course, we sometimes study what the great dead philosophers said. We'd be stupid not to look at what insights they had to offer. But sooner or later, we try to contribute something to philosophical enquiry, and don't merely report on it.
Isn't everyone a philosopher?
It's certainly true that at some point in life everyone comes to ask a philosophical question or two. But it's odd to think that this makes everyone a philosopher, in just the same way that it would be odd to think that running for the bus makes you an athlete. As the term is usually used, the term "philosopher" refers to people who do philosophy for a living. That seems like a sensible way to use the term.
I heard that contemporary philosophers are no longer interested in the big questions, but instead concern themselves solely with the meaning of words?
This is a bad caricature of a view which is itself no longer popular. Philosophers today, like those before now, are interested in a variety of issues, such as the existence of God, how we should live our lives, and how much we can really know about the world. These aren't purely terminological issues.
Hasn't philosophy been superseded by psychology?
No. It is true that many of the questions we ask are questions about people: What can we know? How should we act? What am I? But these don't seem like questions solely about our psychology, and even if they were, they aren't the only questions we address.
Isn't philosophy just a load of rubbish?
Funnily enough, no. Those who say this sort of thing tend to either have no idea what philosophy is, or else are themselves just presupposing certain naive and problematic philosophical theories about the nature of reality and our access to it.
Why didn't I know any of this?
Because philosophy isn't normally taught in schools. Its closest academic cousin is probably maths, but it's hard for anyone to extrapolate from their experience of maths to an idea of what philosophy is like. (On the plus side, subscription to A-level philosophy is on the increase.)
What's the point?
Like all research, part of the point is that we don't yet know what the point will be. But we can point to past advances philosophy contributed to: the rise of the sciences, including more recently psychology, the rise of atheism, the computer, and liberalism. Measuring the exact influence of philosophy of difficult, but it would be silly to infer from this that it has no influence or value at all. Understanding our place in the universe is important to us, regardless of how easy it is to put a price tag on such an endeavour.