What is philosophy?

Philosophers tend to address very abstract questions, such as whether there's a God, whether we have free will, and whether anything is objectively true. More generally, the central topics philosophers are interested in are:

Philosophers try to understand these issues by carefully defining different theories about them, and by trying to demonstrate that some of those theories are superior to others.

How does philosophy relate to science?

In some respects, philosophy is like science. Like scientists, philosophers try to offer theories that make sense of the world. And in many places, the boundaries between philosophy and science are very hard to define. Indeed, until recently, the two were barely distinguished at all: the split is in part the result of the modern university system which has a habit of enforcing strict boundaries between disciplines.

Still, there is a broad difference in methodology. Scientists use logic and reason about things, but they also rely a lot on sensory evidence. Philosophers obviously want their theories to be consistent with the sensory evidence (in turn, with science), but such considerations don't always play a pivotal role: Often, to decide between philosophical theories, you need to rely to a greater degree on the use of logic and abstract reasoning. Abstract questions are often answered by abstract reasoning rather than by minute observation.

So philosophy is similar to maths?

In methodology, yes. Both subjects investigate the world by abstract rational enquiry. The main difference between the two is their subject matter, though even here there is some overlap.

I thought all philosophers died long ago? Don't you just study what they said?

No, and no. There are plenty of professional philosophers around today, and we don't merely report on what the great dead philosophers said. We contribute to the debates they started, or else investigate questions they didn't even consider. Of course, we sometimes study the great dead philosophers, but we read them in order to build their insights into our own theories. Our present theories demonstrably improve on theirs, and will continue to improve as we continue to work on these issues.

Aren't contemporary philosophers no longer interested in the big questions, but instead concerned solely with the meaning of words?

This is a bad caricature of one view within philosophy, and a view that is no longer popular at that. Philosophers today, like those before now, are interested in a wide 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 verbal issues.

Still, philosophers are often careful with language. Partly, this is because the issues we tackle are difficult, and progress is hard if people express themselves imprecisely. But it is also because our choice of language reflects the underlying concepts we are employing, and thereby the subject matter at stake. For example, if we ask whether insects are conscious, a lot might hang on exactly what we mean by "conscious". (See also Construct validity.)