Muggles' Guide to Harry Potter/Magic/Time-Turner

Overview
A Time-Turner in appearance is a small, sparkly hourglass, often worn as a pendant.

Extended Description
A Time-Turner allows for limited time travel; turning the Time-Turner over once causes everyone within the attached chain to go one hour back in time. It also appears that it allows for some control over where in space they will appear. In Harry Potter and the Prisoner of Azkaban, Hermione uses this extensively to attend more classes and exams than would be physically possible; for instance, we see on her first day at school that she is taking three classes at the same time. The Time-Turner is loaned to her by Professor McGonagall under the conditions that she use it strictly for attending classes, and that she never allow herself to be seen using it. Hermione does somewhat break this injunction at the end of the book, when she and Harry use the Time-Turner to revert to a time three hours earlier so that they can save Sirius Black. At the end of that book, Hermione drops a class, which she says will put her onto a schedule that is at least humanly possible; she returns the Time-Turner to Professor McGonagall.

In Harry Potter and the Order of the Phoenix, part of the battle at the Department of Mysteries takes place in a room devoted to the mystery of Time, and Neville, aiming a Stun jinx at a Death Eater, misses when the Death Eater ducks. The spell hits a glass-fronted cabinet containing a large number of hourglasses, which Harry recognizes as Time-Turners. The cabinet falls over and smashes, and then immediately returns to its prior state, only to fall and smash again and again.

In Harry Potter and the Half-Blood Prince, Hagrid, resigned to the fact that none of the Trio are taking his course, mentions that he knows their schedules are full, and reckons that they might have been able to fit Care of Magical Creatures in if they had Time-Turners. Hermione tells him that the entire collection of Time-Turners at the Ministry were destroyed, and that there are none left in the world as far as anyone knows.

Analysis
The Time-Turner is an incredibly useful device; even with the need for precautions, it would be amazingly useful to have the ability to go back and do over the past. The Time-Turner is a device that could destroy any mystery, and make anything possible: with a Time-Turner, Voldemort could "fold" himself, and three of him could face one Harry, thus making his final victory assured. Presumably, it is because it is possible to misuse this device that the author felt they should be destroyed in the battle at the Department of Mysteries.

In Harry Potter and the Deathly Hallows, Dumbledore, speaking of a different event, mentions that something is very deep magic and not well understood by anyone. This implies that there are other areas of magic that are extremely obscure. We could safely expect one of those to be time, as an entire subdepartment at the Ministry is devoted to it. It is entirely possible that the witches or wizards who created Time-turners initially were unable to pass on the secret of how they did so; likely they were naturally adept at manipulation of time, and were unable to pass on their knowledge because no other adept was then available. So while to a Muggle observer, it seems obvious that someone should have just made more Time-Turners, it is quite possible that no living witch or wizard knew how.