The True Cloak of Invisibility. One of the most powerful magical artefacts in existence, it hides its wearer perfectly, every time, even from Death itself.

... But not from Moody's eye, or Hominem Revelio, or the Marauders' Map?

But perhaps simply shutting down all efforts to discovery would not be helpful to the wearer. If Harry had been hidden from Dumbledore in Philosopher's Stone, he might have wasted away before the Mirror of Erised. If Lupin hadn't seen through the Cloak with the Map, he never would have realised Sirius' innocence. If Moody hadn't been able to see through it, Harry would never have recovered the Map. If the Caterwauling Charm hadn't gone off, the Trio would never have infiltrated Hogwarts.

However, when push comes to shove, the Cloak lives up to its legend. It does not respond to Summoning, and Voldemort himself could not detect Harry under the Cloak, even whilst walking right past him.

It is my theory that the Cloak is both semi-intelligent and mildly precognitive. It can sense if being discovered would be beneficial to its wearer, and choose to reveal him or her. The Cloak can't influence the world too much, but it can do its best to steer the course of events in its wearer's favour simply by making itself easier or harder to penetrate.