The Wizard's Wand - A Guide to Crafting and Using Your Own Wand

Wand Cores

Unicorn

-Produces the most consistent magic
-Least subject to fluctuations and blockages
-Generally the most difficult to turn to Dark Arts
-Most faithful, remain strongly attached to their first owners, no matter what they accomplished.
-They don't make the most powerful wands (the wood can make up for it)
-Prone to melancholy if mishandled, hair may "die" and need replacing

Dragon

-Provide wands with the most power
-Capable of the most flamboyant spells
-Learn very quickly
-Can change allegiance if won from first master.
-Bond strongly with current owner
-Easiest to turn to Dark Arts, but will not go that way of it's own accord
-Most prone to accidents
-Somewhat temperamental

Phoenix

-Rarest core type
-Capable of the greatest range of magic, though they may take a bit of patience
-Act on their own accord
-Very picky when it comes to potential owners
-Hardest wands to tame and personalize
-Allegiance is hard won

Summarized from:
https://www.pottermore.com/writing-by-jk-rowling/wand-cores

Of course, these three cores aren't the only ones.

But if you are planning on buying a wand from Ollivander's, these are the only cores we use.

For your education, here are a few more known cores.

The known cores are as listed:
-Phoenix Feather
-Dragon Heartstring
-Unicorn Hair
-Veela Hair
-Thestral Tail Hair
-Kneazel Whiskers
-Troll Whiskers
-Kelpie Hair