Wands represent individual personality and character. Each magically talented person in the book has a wand tailored to fit his or her personality, and each wizard or witch's wand is an extension of the self. Remember how Harry spends so long in Mr. Ollivander's shop, trying to find the perfect wand? That's because every wizard needs a wand that meshes with how he/she works in order to produce the best possible magic. It's harder to do magic with someone else's wand, and even harder to do magic without any wand at all.
Each wand in Ollivander's shop is completely individual, made with very specific magic elements, each resulting in a unique combination. For example, Harry's wand is holly and phoenix feather (phoenixes symbolize rebirth, which could show that he – and Voldemort – is hard to kill). Rowling herself explains:
"[H]olly has certain connotations that were perfect for Harry, particularly when contrasted with the traditional associations of yew, from which Voldemort's wand is made. European tradition has it that the holly tree (the name comes from 'holy') repels evil, while yew, which can achieve astonishing longevity (there are British yew trees over two thousand years old), can symbolise both death and resurrection; the sap is also poisonous." (source)
The fact that Harry chooses a wand with phoenix feather core in it shows how his path is already tied to Voldemort's, since Voldemort also chose a wand with the same core (or rather, the same wand also chose him).
The Sorting Hat
All students entering Hogwarts must try on the Sorting Hat. To the new first-years, it seems like it holds the potential for a trying ordeal – the idea of being "Sorted" carries with it all kinds of creepy, science-fiction implications. Ron, and his fellow first-years, is relieved when they find out that all they have to do is try on the Hat, but Harry is still uneasy. While waiting to be Sorted Harry thinks,
[T]rying on the hat was a lot better than having to do a spell, but he did wish they could have tried it on without everyone watching. The hat seemed to be asking rather a lot; Harry didn't feel brave or quick-witted or any of it at the moment. (7.36)
The Hat tells them, "Try me on and I will tell you / Where you ought to be" (7.33). In other words, after putting on the Hat, each student will know the house to which he or she belongs. While this seems true for all the other first-years, who put on the Hat and hear it yell out different house names, it's a little different in Harry's case. The Hat says it will determine which House each student "ought to be" in, but it offers Harry a choice. When Harry thinks at the Hat, "Not Slytherin, not Slytherin" (7.68), the Hat responds,
"Are you sure? You could be great, you know, it's all here in your head, and Slytherin will help you on the way to greatness, no doubt about that – no? Well, if you're sure – better be GRYFFINDOR!" (7.69)
Here, the student's preference for one house over another is just as significant as the personality traits that qualify him for those houses.
Want to know what house you belong in. Take this quiz! Or maybe this quiz. Heck, you might as well take this quiz too!
The Sorcerer's Stone
The Sorcerer's Stone is a mythical-sounding object that does two things: 1) transforms base metals into gold; 2) creates an immortality potion (the Elixir of Life). Here's what Hermione learns about the Stone from a book from the Hogwarts library:
"The ancient study of alchemy is concerned with making the Sorcerer's Stone, a legendary substance with astonishing powers. The stone will transform any metal into pure gold. It also produces the Elixir of Life, which will make the drinker immortal." (13.50)
But alchemy isn't something that Rowling made up; the idea of Nicholas Flamel using the Sorcerer's Stone (more often called the "Philosopher's Stone") is actually based in legend and fact. Here's what Rowling has to say about Flamel on her website:
Nicholas Flamel is a historical character. Flamel lived in France in the fourteenth century and is supposed to have discovered how to make a philosopher's stone. There are mentions of sightings of him through the centuries because he was supposed to have gained immortality. There are still streets named after Flamel and his wife Perenelle in Paris. (source)
In Harry Potter and the Sorcerer's Stone, the characters aren't nearly as interested in the Stone's potential for making gold as they are in its capacity to bestow eternal life. Most specifically, Voldemort desires the Stone as a means of restoring himself to his former glory – using the Stone, it seems, has far less terrible side effects than drinking unicorn blood. As Dumbledore points out, however, both riches and immortality cause nothing but problems:
"You know, the Stone was really not such a wonderful thing. As much money and life as you could want! The two things most human beings would choose above all – the trouble is, humans do have a knack of choosing precisely those things that are worst for them." (17.107)