"A jet of green light issued from Voldemort's wand just as a jet of red light blasted from Harry's - they met in midair - and suddenly Harry's wand was vibrating as though an electric charge were surging through it; his hand seized up around it; he couldn't have released it if he'd wanted to - and a narrow beam of light connected the two wands, neither red nor green, but bright, deep gold." - Goblet of Fire
The sequence that followed was extraordinary. Both Harry and Voldemort lifted into the air and were moved by an unknown force out of the graveyard. Threads of light passed over and around the two until a dome-like exterior appeared. Then Harry heard, emanating from the light, a phoenix's song. It gave Harry hope in the following battle of the minds that ensued between he and Voldemort moments after. Finally, after Voldemort lost the preliminary scuffle, the force of the Priori Inacantatem was pushed on Voldemort and his own wand. Out of which appeared the ghosts of those he had killed in reverse order.
(And that is why Priori Incantatem also goes by the name of "Reverse Spell Effect". That is exactly what occurs: spells are regurgitated in ghost-like form.)
It was a miraculous event which spared Harry's life. A chance, among quite a few in the series, for Harry to see his parents. Even Ollivander, a master of wands and a scholar of wandlore, was amazed at the effects of the Priori Incantatem between Voldemort's and Harry's wand. Throughout the series, he did well in guessing the nature of this mystery; but the fact remains, it was guess-work. There might still be more to suss out
Here's what we know about Priori Incantatem:
1) It occurs when wands with "twin cores" (or cores from the exact same magical creature) are forced to battle each other.
2) The connection of two projectile spells cuminates in a golden string of energy which glues the wands together as well as the carriers.
3) The effects from the spell priori incantanto are applied to one of the two holders' wand; the one with the weaker will.
Wands seem to resonate with their carriers; that much is clear. And yet, in the scene where Voldemort chases Harry on the motorbike, Harry's wand acts on its own to destroy Voldemort's substitute with strong golden flames Harry had never used before. Ollivander concluded that Harry's wand had somehow detected the intruder wand and borrowed strength from its "brother" to destroy the outsider. However, is it not also conceivable that the wand acted the way it did because it resonated with Voldemort?
From the described event, one would think the wands knew something their masters did not. They may of had some form of boundless communication, which could only be described by the boundless possibilities of magic. And yet, these wands must constantly face off in the impossible task of destroying the other which is (as described) impossible.
Taking into consideration a wand's shifting loyalty, the ways in which they mimic their masters, and their ability to possibly communicate!... Can someone say without a doubt that they do not think?