Fidelius Charm

"An immensely complex spell involving the magical concealment of a secret inside a single, living soul. The information is hidden inside the chosen person, or Secret Keeper, and is henceforth impossible to find — unless, of course, the Secret Keeper chooses to divulge it. As long as the Secret Keeper refused to speak, You-Know-Who could search the village where Lily and James were staying for years and never find them, not even if he had his nose pressed against their sitting room window!"

- Filius Flitwick explaining the charm

The Fidelius Charm (incantation unknown) is an extremely difficult, multifaceted and potent charm that can be used to conceal a secret inside an individual's soul; the witch or wizard who houses the secret is known as the Secret Keeper. A dwelling whose location has been protected by this spell is then invisible, intangible, unplottable and soundproof. This is an extremely old spell, one of the most ancient of all.

Becoming Secret Keeper
When the charm is initially cast, one can choose more than one person to be Secret Keeper, though it is unknown if there is a limit as to how many people can carry the same secret in this way. Once the Secret Keeper dies, all the people who have learnt the secret from them will become the new Secret Keepers. If the Keeper has chosen not to reveal the secret to anyone (so that they are the only Keepers) then the secret rests with them forever.

It is unknown what criteria must be met before someone can qualify as a secret-keeper. However, it appears that, to be eligible, one must be capable of divulging the secret if they wish to, as this would explain why wizards do not simply bestow the secret onto ceatures who cannot speak a human language, or creatures such as house elves, who can simply be ordered by their masters to not divulge the secret (as such an order would override their free will).

Revealing the secret
Once the secret has been implanted, the only method for another to learn it is for the primary Secret Keeper to divulge it to that person directly, it cannot be discovered any other way. Additionally, it appears as if the disclosure must be voluntary (as evidenced by the fact that no dark wizard is ever seen attempting to coerce the disclosure by means of Veritaserum, Legilimency or the Imperius Curse). The people who the Secret Keeper has already divulged the information to (secondary Keepers) would not be able to pass it along even if they wanted to and the information in question can not be learned even by stumbling across it, even if one were staring the secret in the face. For example, if someone were to hide a location in this way, even if a person were to stumble into the specific area they would not be able to recognise it for what it was (i.e. as the area they were looking for) and they would not therefore learn the location accidentally. The method for the primary Keeper to voluntarily tell another person the secret can be verbal or written. It is also possible that primary Keepers can reveal information to another by showing it to them. This theory is postulated because whilst 12 Grimmauld Place was under this charm and being used as a base of operations for Harry, Hermione and Ron only after Dumbledore's death, the Death Eaters evidently became aware of 12 Grimmauld Place's approximate location knowing that Harry Potter had inherited the property from Sirius Black. Despite watching the gap between numbers 11 and 13 when on the lookout for Harry, the house remained invisible to them until Yaxley grabbed onto Hermione Granger's arm while she, Harry, and Ron Weasley Disapparated away from the Ministry of Magic to 12 Grimmauld Place, where they had been hiding. It is explained that because Hermione (along with everyone else who knew the house's location) had been rendered Secret-Keeper by Dumbledore's death, she inadvertently took Yaxley past the protections of the Fidelius Charm. The trio subsequently abandoned the house as a hideout.

Once the secondary Keeper has received the information in either of these ways the information will still appear hidden to them until they start thinking about it, at which point the secret will reveal itself to them alone. For example, Alastor Moody gave Harry Potter a parchment with Albus Dumbledore instructing where the Order of the Phoenix's headquarters is. He then merely has to think about the said location whilst in the correct area and it would appear out of thin air, allowing him entry.

Etymology
"Fidelis" is Latin, meaning trustworthy and faithful. "Fidelius" is the neuter comparative form of this word, meaning more trustworthy and more faithful. "Fidelius" can also translate to authors.

Behind the scenes

 * contained a few notable discrepancies regarding the Fidelius Charm. For example, it is not known whether or not Yaxley had learned the location of 12 Grimmauld Place by "piggybacking" on Hermione, or even if he had comprehended where he had been taken by Hermione; he may simply have stumbled out of the hideout unaware of his true location between exiting and entering it.
 * When the Weasley family was forced into hiding, Bill told them that Arthur was Secret-Keeper and also told them that they were staying at Aunt Muriel's. Since Arthur was Secret-Keeper, it would logically follow that Bill should not have been able to tell the trio about the Weasleys' whereabouts. Also, when Harry, Ron, and Hermione were captured and taken to Malfoy Manor, Ron told Dobby where to take the hostages when the house-elf came to rescue them. There are several theories to explain these incidences: whilst it is possible that Bill and Arthur were both Secret-Keepers for Muriel's, and that Ron had been made into an additional Secret-Keeper for Shell Cottage, this was never specified. It is more likely that the Fidelius charms were cast after the trio arrived at the cottage, since Bill and Arthur would not have known that they were in immediate danger until after the first group of escaped prisoners arrived at Shell Cottage. Or perhaps it is due to the fact that elf magic is beyond that of wizard magic (i.e. being able to apparate in/out of Hogwarts, being able to apparate in/out of Malfoy Manor). However if house elves can bring people into protected places without permission, any one can ask a house elf to break into any places if he owns that house elf.
 * It is also conceivable that while anyone can know where people and places protected by a Fidelius charm are located, once the charm is cast, only the secret keeper can reveal, as in, make detectable, the location. This is backed up by the fact that Death Eaters knew that number 12 Grimmauld Place existed, and were stationed outside it, even though it was not visible to them. This alternate understanding of the word reveal makes it possible to understand a number of these inconsistencies, such as why Dumbledore immediately knew where Hagrid should look for Harry after his parents' deaths, even though Dumbledore had not been told the location of the Potters' house by its Secret Keeper.
 * Also, the Fidelius Charm appears to be impervious to a Taboo Jinx, such as the one placed on Lord Voldemort's name after the Fall of the Ministry of Magic. Several times, unaware of the Taboo, Harry and Hermione say Voldemort's name without the Fidelius Charm being broken on 12 Grimmauld Place or even the Death Eaters appearing to be aware that the Taboo was crossed. This directly contrasts the result of Harry saying the name while in the tent, which was followed immediately by the enchantments and charms placed there disappearing and a team of Snatchers Apparating outside. The Fidelius Charm seems, therefore, to not only render those residing in the charmed location as undetectable as the location itself, but does not transmit information which could counter the charm (such as speaking a Taboo name) outside its sphere of influence.
 * It is also worth noting that, when Dumbledore died, Snape would have become a primary Secret-Keeper. Therefore, he should have theoretically been able to tell the Death Eaters the location of Grimmauld Place. However, it was not until Hermione accidentally leading Yaxley to the place did the Death Eaters (possibly) discover it. Snape either must've chosen not to reveal the location because of his secret loyalty to the Order, or may have been unable to reveal the location because of the Tongue-Tying Curse cast by Alastor Moody. Neither of these theories are ideal, though, since Voldemort would likely have tortured, or even killed, Snape had he declined to reveal the location, and if a tongue-tying curse was used to keep Snape from talking, that would kill the theory mentioned above that, to be secret-keeper, one must be capable of revealing the secret if they choose to (which, in turn, would beg the question as to why only humans, and not house elves or common animals, are never chosen as secret-keepers).
 * Rowling once stated that, should the Secret Keeper die, anyone who knew the secret would be permanently unable to pass it on . Obviously, this is no longer the case.

Notes and references
Encantamiento Fidelio Sortilège de Fidelitas Uskollisuusloitsu Incanto Fidelius Фиделиус Zaklęcie Fideliusa