In JK Rowling's seminal Harry Potter books, many wands are described in detail and many of the attributes assigned to each reflects its owner in some subtle (and some less so subtle) ways. The author studied the ancient Celtic art of tree signs to influence the choices of wand woods and decided upon her own choice of powerful magical cores so that specific combinations would carry meaning.
Take for example the twin cores of Harry's and Voldemort's wands, both coming from the tail feathers of Fawkes the Phoenix. And the importance doesn't stop there. Look at their wand woods: Harry's is made of holly, with clear connections to the traditional Christian views of right and wrong and the idea that the wood is protective like Harry himself. Voldemort's, on the other hand, is constructed from yew, a tree with several deep-rooted connotations with death.
In the first two Harry Potter films, each wand was subtly different from the others, but the designs were simple and it could be difficult to tell them apart. It seemed, fairly enough, that the books would have to hold the unique characters of the wands, and that these details simply couldn't be translated to the screen.
But then along came Alfonso Cuarón to direct The Prisoner of Azkaban. He gave each principal actor a chance to choose their own wand from a more visually dynamic range. As concept designer Adam Brockbank explained, "their wands are the sheerest expressions of their characters."