Like the Elder Wand, the Resurrection Stone is one of the "Hallows" mentioned in the "Tale of the Three Brothers." (Remember that from the previous installment of HP and the Deathly Hallows?)
And, also like the Elder Wand, the Stone possesses some pretty crazy powers…but with a downside.
If you were paying attention during the last film, you know that people can use the Stone to talk to the dead and keep them around you in ghost form. Sound pretty good? Well, sadly, bringing back the dead can actually be pretty depressing—since, you know, these ghostly folks can't actually come back and live with you fully.
So, in the tale, the brother who owned the Stone ended up killing himself to be with his former love for real, rather than have her around in this weird "halfway" sense. The Stone's power really just reminded the dude and his ghost love of what they were missing, rather than making them feel like they were together again.
Dumbledore left Harry the Resurrection Stone (disguised in a Snitch) in his will, and Harry uses it just before he goes to battle Voldemort. However, he doesn't use it to try to bring the dead back for an extended period or anything. Rather, he just wants to have a chat with his parents and some other departed friends (like Sirius and Lupin) to give him strength before he himself goes to accept death.
See, this is the remarkable thing: Harry totally embraces the fact that he thinks he's going to die, which he signals to the audience by dropping the Stone on the ground before going to face Voldemort.
Now, you'd think that he might want to keep something called the Resurrection Stone around, just in case it might help prevent him from being killed, but Harry is totally willing to sacrifice himself if it means the end of Voldemort.
In short, his abandoning of the Stone shows just how brave he really is.
Of course, irony of ironies, the Stone may have ended up protecting Harry after all, in some way. We already knew Harry inherited the invisibility cloak from his father, and we learn at the end of the film that the Elder Wand's allegiance was actually to him.
Sure, he dropped the Stone off before facing Voldemort, but he was still the owner, right? So, technically, Harry owned all three objects and therefore (according to the myth) would have been the master of death.
Is that why he survives Voldemort's Killing Curse again? Who knows? But the Stone definitely symbolizes Harry's willingness to lose it all and embrace death, which ends up saving everyone from Voldemort—even Harry himself.