From Pottermore's Wand Cores page (it has to be unlocked), as an quote from Ollivander:
After much experimentation and research, I concluded that only three
substances produce wands of the quality to which I am happy to give
the illustrious name of Ollivander: unicorn hair, dragon heartstring
and phoenix feather. Each of these costly and rare materials has its
own distinct properties.
He called these the "Supreme Cores" and then describes the various properties of each of the three:
Unicorn
Unicorn hair generally produces the most consistent magic,
and is least subject to fluctuations and blockages. Wands with unicorn
cores are generally the most difficult to turn to the Dark Arts. They
are the most faithful of all wands, and usually remain strongly
attached to their first owner, irrespective of whether he or she was
an accomplished witch or wizard. Minor disadvantages of unicorn hair
are that they do not make the most powerful wands (although the wand
wood may compensate) and that they are prone to melancholy if
seriously mishandled, meaning that the hair may 'die' and need
replacing.
Dragon
As a rule, dragon heartstrings produce wands
with the most power, and which are capable of the most flamboyant
spells. Dragon wands tend to learn more quickly than other types.
While they can change allegiance if won from their original master,
they always bond strongly with the current owner. The dragon wand
tends to be easiest to turn to the Dark Arts, though it will not
incline that way of its own accord. It is also the most prone of the
three cores to accidents, being somewhat temperamental.
Phoenix
This is the rarest core type. Phoenix feathers are capable of the
greatest range of magic, though they may take longer than either
unicorn or dragon cores to reveal this. They show the most initiative,
sometimes acting of their own accord, a quality that many witches and
wizards dislike. Phoenix feather wands are always the pickiest when
it comes to potential owners, for the creature from which they are
taken is one of the most independent and detached in the world. These
wands are the hardest to tame and to personalise, and their allegiance
is usually hard won.
Alternative source here