This is an old one. The Greek historian, Plutarch, wrote that the Athenians kept the hero Theseus' ship on display in their city. As the wooden timbers became rotten, the Athenians would preserve it by pulling the timbers out and replacing them with new ones. But were they really preserving it? If you replace every single timber in the ship, is it the same ship? Suppose the Athenians had yanked out the timbers before they even began to rot, and one by one use them to build a new ship. Which one has the best claim to being the original ship?
This is the sort of problem that a lot of people roll their eyes at. However, while some of us may be able to shrug off questions like this without any trouble, philosophers are kept awake at night by the Ship of Theseus. That's because it's a very, very bothersome problem once you apply it to other things.
For example: of all the cells in your body, how many were there a month ago? A year? Ten years? By what authority do you even say that you're the same person you were a decade ago? If the ship in Athens was no longer the Ship of Theseus after a century, then you're not the same person you were ten years ago. Not so irrelevant now, is it? Suddenly, the Ship of Theseus becomes a very personal matter, because each one of us is a real-life case. A question like this forces us to devise more sophisticated definitions of what makes something identical over time.
To further understand why this is so unsettling, we have to look at the history of Western thought. Since Aristotle, our philosophy has been dominated by the idea of substance. A substance, roughly speaking, is a thing which simply is what it is regardless of its relationship to anything else, and regardless of what properties it has. For example, suppose that S is a substance. S could turn blue, and then red, and then green, and change in shape from square to circular to triangular, but S = S no matter what. S just is S by virtue of being S, and not being any other thing. This all sounds like perfectly tautological common sense, which it is, but it suddenly looks questionable when we bring up the Ship of Theseus. Clearly, something has to give.
The first answer to problems of this nature is known as mereological nihilism. Mereological nihilism says that nothing is part of anything else, and the only real things in the universe are 'simples'. In other words, large objects – rocks, trees, people, and so on – aren't even real. Nor are the molecules of which they are supposedly constituted. Nor are the atoms. The only 'real' entities, on this view, are the very smallest pieces of matter, which are the simples spoken of earlier. To say, “I see a person,” on this view, is simply to say “I see a big group of simples arranged person-wise.” Now, there are some genuine philosophical challenges to this view, which I will not go into here. But most people, upon first hearing it, will have trouble accepting it purely because of how counter-intuitive it is. Sure, I could claim to be a mereological nihilist, but it's awfully hard to point to a chair and say, “That chair is not real.”
One view that came about in the early 20th century was called Process Philosophy the brainchild of Alfred North Whitehead. As far as I know, it wasn't conceived of in response to the Ship of Theseus specifically, but it does get around the problem. In a nutshell, Process Philosophy doesn't view the fundamentals of the universe as substances or things: rather, it views all of reality as a dynamic, constantly-changing process. A tree is not a thing which remains the same over time: it's a process which begins as a seed and ends as a log which eventually becomes soil.
The Ship of Theseus remains one of my favorite conversation-starters on long car rides or in new company. Try it out sometime and see what kinds of answers you get.