Since many philosophy courses at university don't require philosophy to have been studied at sixth form, it's possible to pick it and have no idea of what philosophy is about. So this article contains suggestions for books that will give you a flavour of what philosophers do and what philosophy deals with. Feel free to add to it with your suggestions and comments. First draft of this article was based on this thread.
Don't read this article and assume you should read everything: to begin with, hopefully you're doing other interesting things with your life, and you'll also find some of the books have significant overlap.
- The general books in the first section give you, appropriately, a general overview of what sort of things philosophy talks about. Books in the second section talk more about specific areas of philosophy, but should still be readable to an amateur.
- If you start to get the impression that all philosophers are Greek and ancient, or that all philosophy involves funny symbols that look like maths, or that every philosophical text is written in dense prose designed to keep you out, it might be worth exposing yourself to something completely different to shake that feeling off.
- The third section contains texts written by genuine philosophers for genuine philosophers, but leaves out the stuff that's almost impossible to get into without huge effort.
- The last section mentions some places on the internet you might want to go to help understand what you're reading or get a more dynamic view of actual philosophy being done.
General introductions to philosophy
- Julian Baggini, The Pig that Wants to be Eaten (A nice collection of short chapters, each on a different philosophical problem or issue. Easy to dip in and out of.)
- Simon Blackburn, Think: A Compelling Introduction to Philosophy
- Jostein Gaarder, Sophie's World
- Adam Morton, Philosophy in Practice
- Thomas Nagel, What does it all mean?
- Bertrand Russell, The Problems of Philosophy (Easy to understand but sometimes biased)
- Bertrand Rusell, A History of Western Philosophy
- Nigel Warburton, Philosophy: The Basics
- Wikibooks, Introduction to Philosophy, 
- Richard Wilkinson, The Spirit Level
Introductions to specialised areas
- The Very Short Introduction to...
These are generally written by experts, and can be found on a huge range of topics.
- Peter Singer, Practical Ethics
- Simon Blackburn, "Being Good"
- Wilfred Hodges, "Logic"
- Samuel Guttenplan, Languages of Logic
I would recommend everyone interested in philosophy or any subject that will require you to analyse arguments (any humanity, and science, most other things) teach themselves basic logic. It will improve your critical thinking skills no end, and is much less effort than it looks.
Philosophy of mathematics
- Stewart Shapiro, Thinking About Mathematics (A very readable introduction to an area of philosophy that's easily missed if you just follow a sixth-form syllabus or basic introduction to the topic.)
- Jonathan Wolff, An Introduction to Political Philosophy
Key original texts
- Plato, The Republic
- Aristotle, Nichomachean Ethics
- Rene Descartes, Meditations
- Thomas Hobbes, Leviathan
- Baruch Spinoza, Ethics
- David Hume, Treatise on Human Nature
- Immanuel Kant, Critique of Pure Reason
- G.W.F Hegel, The Phenomenology of Spirit
- Karl Marx, The Communist Manifesto
- John Stuart Mill, On Liberty (This book has informed political and ethical discussions ever since it was written, and is a pretty easy read.), "Utilitarianism"
- Friedrich Nietzsche, The Genealogy of Morality
- Martin Heidegger, Being and Time
- John-Paul Sartre, Existentialism and Humanism (Originally based on lecture notes, so more readable than a lot of existentialist philosophy, and a good introduction to Sartre's thinking.)
- Ludwig Wittgenstein, Philosophical Investigations (The foundation of modern analytical philosophy)
Other sources of information
- Philosophy Bro, 
- Rust Belt Philosophy,  (The author takes current writing and disassembles the bad reasoning and argument that comes out of the pens of so many modern-day commenters.)
- Stanford Encyclopedia of Philosophy,  (Like Wikipedia, but written by experts in all fields of philosophy to an academically rigorous standard.)