What does Philosopher mean in Britain? Obviously it's not the American meaning or the book & movie wouldn't have changed the title for the USA.
As per Valorum's comment, the Philosopher's Stone is a legendary alchemical item.
The philosopher's stone, more properly philosophers' stone or stone of the philosophers (Latin: lapis philosophorum) is a legendary alchemical substance capable of turning base metals such as mercury into gold (chrysopoeia, from the Greek χρυσός khrusos, "gold", and ποιεῖν poiēin, "to make") or silver. It is also called the elixir of life, useful for rejuvenation and for achieving immortality; for many centuries, it was the most sought goal in alchemy. The philosophers' stone was the central symbol of the mystical terminology of alchemy, symbolizing perfection at its finest, enlightenment, and heavenly bliss. Efforts to discover the philosophers' stone were known as the Magnum Opus ("Great Work").
It's worth noting that there's some debate as to how literal the "turning lead into gold" goal was, or if alchemy was a more, well, philosophic life approach of self-improvement with turning lead into gold being a metaphor.
Why did they change it? Because the publisher's didn't think kids would understand the reference.
The title of the movie was changed to "Harry Potter and the Sorcerer's Stone" because the book's name was also changed to this. JK Rowling agreed to change the title of the book from "Harry Potter and the Philosopher's Stone" to "Harry Potter and the Sorcerer's Stone" because the publishers thought that American readers wouldn't pick up a book called "Harry Potter and the Philosopher's Stone" because they would associate the word Philosopher with some old, boring guy and not with magic. The word Sorcerer brings magis to mind for American children. (citation)
The word philosopher itself means the same thing (or rather, things) on both sides of the pond. The issue was that the American publisher thought that people would not know what "philosopher's stone" (i.e. the whole phrase) meant. They decided that "sorcerer's stone" would be a better term to use, because even if you've never heard of such a thing, you know it has to do with magic. With "philosopher's stone", if you don't know that it's the alchemists' Holy Grail (and the word "philosopher" itself should really be interpreted as "alchemist"), you can't place it in any context, so you might conclude that it's something to do with actual philosophy. And what child wants to read about philosophy?
In the context of "Philosopher's Stone", a philosopher is a scientist (in medieval usage, a natural scientist -- as opposed to, say, a mathematician) -- specifically, an alchemist. The Philosopher's Stone is the ultimate goal of alchemy, giving the ability to perform any transmutation, as well providing an Elixir of Life that confers immortality.
More broadly, "philosopher" is made up of two Greek roots: "philos" -- love (non-sexual, brotherly), and "sophias" -- wisdom. A philospher, then is someone who loves wisdom. Historically, the primary occupation of philosphers was to engage in reasoned argument with the intention of discerning truth from supposition -- earning their name by determining what was, in fact, "wisdom" (as opposed to supposition, speculation, and so forth).