Harry was not a Horcux in the formal definition of the word but, because he was entirelty unique (having a slice of soul intended for a horcux dwell inside of him for 17 years) it is easy to lump him in with horcuxes for brevity. As several other answers have quoted, Dumbledore was not saying that Harry was a dark magical container for Voldemorts soul fragment that Voldemort used the Horcux spell to create. Nor, was Harry possessed by Voldemort because of the soul fragment. We know this because Harry was possessed by Voldemort in the Fifth book and it was a completely new and terrifying experience for him (and Voldemort for that matter).
Instead, like a Horcux Harry possessed a fragment of Voldemort's soul that would teather Voldemort to life while still in Harry and other than the fact that if Harry had died Voldemort's soul fragment would also have been distoryed do to lack of container--that is where the similarities stop. Harry experiences visions of Voldemort's emotions, and circumastances but through the lense of his own brain, morals and personality and was never swayed to agreeing with the values, actions or emotions of Voldemort's soul fragment. There is a spot where he didn't even realse he was feeling elation from Voldemort's soul connection because he himself was so miserable. Harry felt the feeling from the soul tie but it didn't effect his misery at all (so much so he couldn't even recogize the physicial/mental symptoms as happiness until much later). He didn't have a negative effect on those around him. There is a theory that him being a Horcux(which he is not) is why The Durselys were so awful but it doesn't hold any water as Harrys presence doesn't effect his dormates, the Weasleys, Sirus, or Ron/Hermione when they stay /live together at various points in the books. The fragment of soul has the opposite effect in fact, it intergrated into Harry's mind body and soul adding skills, aptitudes to his innate gifting as well as insight to know and understand his greatest enemy that no other person would have ever known because Voldemort was so private and secretive. So much so that Dumbledore says Voldemort's closest followers would give anything to have Harry's insight into their masters private world. This point (the souls intergration into him rather than him being tainted by it presence) is proven by his ablity to produce a patronus even with a piece of the evilest soul in history cohabitating with his soul. Patronuses are not only hard magic but only people with pure hearts are able to cast them and by pure 'heart' I think they are actually referring to whole souls.
The biggest thing that should leave us without a shadow of a doubt that Harry isn't a true Horcux is that he was able to die for the people in the hogwarts battle and invoke the deep love magic that serves as magic protection against the killing curse/evil magic. If Harry was a horcux which is intrinsically bad, dark magic- evil he would have be unable to access the opposite power magically- love. His horcuxness would have inhibited it, instead it cleansed him of the piece of soul that never belonged, thus the power of love always being stronger and more far reaching than the powers of darkness and evil, in ways that evil cannot even concieve or imagine.