Are the staves that the Istari wield made of anything unique or are they made from normal wood and given magical properties by their user?
The only place in LotR where any reference is made to what a wizard's staff may be made of is Frodo's poem in Lorien:
an old man in a battered hat
who leaned upon a thorny staff
"Thorny" here certainly suggests that the staff is of wood.
It is wrong, I think, to imagine a wizard's staff as having specific magical properties. True, there are many instances in LotR where it may seem so: Gandalf's lighting of wood on Caradhras ("he thrust the end of his staff into the midst of it"), his entry into Meduseld ("The staff in the hand of a wizard may be more than a prop for age"), and, of course, "Saruman, your staff is broken" are just some examples. Gandalf's demand that "you will first surrender to me the Key of Orthanc, and your staff" is also suggestive of the same.
However, that's just not the way things work in Tolkien. Gandalf's staff was broken on the Bridge of Khazad Dum but yet he managed to fight and kill a Balrog:
Thunder they heard, and lightning, they said, smote upon Celebdil, and leaped back broken into tongues of fire.
This was almost certainly a description of Gandalf's own magic; compare with the description of the Nazgul's attack on him on Weathertop:
As Frodo lay, tired but unable to close his eyes, it seemed to him that far away there came a light in the eastern sky: it flashed and faded many times. It was not the dawn, for that was still some hours off.
'What is the light?' he said to Strider, who had risen, and was standing, gazing ahead into the night.
'I do not know,' Strider answered. 'It is too distant to make out. It is like lightning that leaps up from the hill-tops.'
And to confirm that lightning-type effects are a sure sign of it being Gandalf, let's look at what happens when he rescues Faramir:
...it seemed to Pippin that he raised his hand, and from it a shaft of white light stabbed upwards.
Based on this, it's definite that Gandalf doesn't actually need his staff for magic; in the Khazad Dum example he doesn't have it but yet achieves much the same effect.
So now we get into speculation, and what I'm going to speculate is that a wizard's staff in Middle-earth is not a D&D-type "magic item" but rather a sign of authority, a symbolic prop. By breaking Saruman's staff Gandalf actually did far more; he disconnected him from his origins in Valinor, he cast him out from the Order, and the breaking of the staff was just an outwardly visible manifestation of that.
Of course there is still a lot left unanswered. Why did Gandalf thrust his staff into the wood on Caradhras? Don't know. Why did Gandalf ask for Saruman's staff before breaking it? Don't know. Why did Saruman taunt Gandalf about wanting "the rods of the Five Wizards"? Don't know. In the end this is just a matter that Tolkien left unclear and unanswered.