Wands
First off, you need your wand to do magic or to poke someone in the eye if you haven't learn any spells yet. Either way, you need one. So the admins here at Duplicity have decided to break it down for you so that it's not so random. Don't forget the wand chooses the wizard...
Wood
Wands choose their wizard according to birthdate. So take a look at these cutoffs to find out what kind of wood you'd have in your wand.
December 24 - January 20 = Birch
January 21 - February 17 = Rowan
February 18 - March 17 = Ash
March 18 - April 14 = Alder
April 15 - May 12 = Willow
May 13 - June 9 = Hawthorn
June 10 - July 7 = Oak
July 8 - August 4 = Holly
August 5 - September 1 = Hazel
September 2 - September 29 = Cherry
September 30 - October 27 = Ivy
October 28 - November 24 = Reed
November 25 - December 23 = Elder
Elements
So now comes the core to your wand. A very crucial part, otherwise doing magic would be a little difficult lacking a magical ingredient in the wood. So, without further adieu lets move to the wood cores.
Unicorn Hair - Often in a woman's wand, the unicorn hair is quite good for charms.
Dragon Heartstring - Very good for transfiguration, it is usually found in a whippy wand.
Runespoor Fang - This is a wand used for the dark arts, or defence against it. It often makes for a stubborn and rigid wand.
Phoenix Feather - Very rare as few pheonixs will willingly or unwillingly give a feather to a wand maker. These wands are always quite expensive because of the rarity of the core. This wand is usually good for those pure of heart.
Leprachaun Hair - This, while rare is quite inexpensive. Unusually good for illusion charms or disillusionment charms.
Chimaera Scale - These are very good at transfiguration because of the mixed creature they come from. While not rare they can be expensive.
A Side Note
The above items are just examples, pretty much anything with magical properties can be put into a wand.