An Invisibility Cloak is a cloak which renders its wearer invisible to the naked eye. They are incredibly rare (PS12). There are several types of Invisibility Cloaks: some are merely cloaks spelled with a Disillusionment Charm; others are imbued with a Bedazzling Hex; still others are woven from Demiguise hair, which eventually lose their ability until they become opaque. One Invisibility Cloak, however, is a true Cloak of Invisibility, which truly makes its wearer invisible, regardless of what spells are cast at it, and endures eternally. This last cloak is the third of the Deathly Hallows, objects that purportedly make any who unites all three of them the Master of Death, created by the Peverell brothers. Ignotus Peverell created the Invisibility Cloak (DH21, DH35).
Albus Dumbledore and other skilled wizards—such as Lord Voldemort (DH24) and Grindelwald (DH35)—have no need of an Invisibility Cloak to make themselves invisible; a powerful Disillusionment Charm is enough to achieve that. Albus frequently uses the Homenum revelio to detect human presence (BLC), which allows him to notice Harry's presence even under the Cloak (PS12).
Ignotus Peverell's Invisibility Cloak is a silvery, light cloak which feels odd to the touch, like water woven into cloth (PS12). It is represented in the symbol of the Hallows by a triangle circumscribed about a circle split by a vertical line (DH21).
A fantasized version of the creation of the Cloak is found in The Tale of the Three Brothers, a story by Beedle the Bard which tells the story of the Hallows. According to legend, a man who had cheated Death asked for something that would keep him out of Death's sight, and Death gave him his own Invisibility Cloak. The man then wore the Cloak for the rest of his life, taking it off when he was old and ready to meet his death (DH21).
After Ignotus' death, the Cloak passed down through his descendents, ending up with the Potters. Harry inherited the Cloak from his father, receiving it as a gift from Albus Dumbledore for Christmas in 1991 (PS12). Dumbledore had been in possession of the Cloak when the Potters died because, upon hearing of the Cloak from James, he realised that the Cloak was the third of the Hallows, and he wanted to examine it (DH35). Dumbledore took it before the Potters truly went into hiding—before July 31, 1981—and its absence contributed to James' feeling a little stir-crazy at being cooped up inside his house all day (DH10).
Harry used the Cloak for several purposes ever since then, using it to sneak around the school, hide from Death Eaters, and in general stay safe; he did not realise what it was, or that it was in any way special, until he visited Xenophilius Lovegood, who told the Trio of the Deathly Hallows and perfectly described the Cloak (DH21).
Though Harry united all three Deathly Hallows, becoming Master of Death, he only chose to keep the Cloak, as it was the safest and most useful of the Hallows—and it was rightfully his, as Ignotus' descendant (DH36).