There are many different types of wood so I will list the ones found on Pottermore and go into detail on three.
Acacia
Alder
Apple
Ash
Aspen
Beech
Blackthorn
Black Walnut
Cedar
Cherry
Chestnut
Cypress
Dogwood
Ebony
Elder
Elm
English Oak
Fir
Hawthorn
Hazel
Holly
Hornbeam
Larch
Laurel
Maple
Pear
Pine
Poplar
Red Oak
Redwood
Rowan
Silver lime
Spruce
Sycamore
Vine
Walnut
Willow
Yes

I will now go into detail about three of these; Elder, Holly and Fir. Elder, because it was Dumbledore's. Holly, because it was Harry Potter's and Fir, because that's mine.
Elder is the rarest of all wand woods. It is also one of the trickiest. It spurns all but the most superior of a given group. Holly is a protectors wand. It works well for someone on a dangerous quest. If paired with a phoenix feather core it becomes a deadly weapon if they find the right witch/wizard to work for. Fir is known as the survivors wand. These wands command staying power and strength of purpose. They work well for the strong minded and sometimes intimidating.