Of the Horcrux, wickedest of magical inventions, we shall not speak nor give direction
- from the introduction of Magick Moste Evile (Harry Potter and the Half Blood Prince, Ch. 18)
Given Voldemort's lack of a magical background and Dumbledore's stringent ban on Horcruxes, we must assume that Voldemort owed his knowledge of the existence of Horcruxes to this passing mention of them in a Hogwarts library book. J K Rowling uses the term "Horcrux" to refer to an object in which a wizard or witch has concealed a part of his/her soul. Whilst the object does not necessarily need to be inanimate, Dumbledore suggests that using a living creature as a Horcrux is risky since the living creature can move and think for itself, independently of the implanted fragment of soul. The purpose of a Horcrux is to protect the soul fragment from anything that might happen to the body of the person who has removed the soul fragment. Whilst the Horcrux is kept safe, that person will continue to exist even if his/her body is badly injured or destroyed.
Slughorn tells the young Voldemort that to create a Horcrux, the witch or wizard has to split his/her soul into fragments by committing a murder. A spell then has to be used to implant the soul fragment into the Horcrux object, but Slughorn does not know the details. (HBP, Ch. 23). We learn that Voldemort became obsessed with avoiding death at a fairly young age (perhaps as a result of his mother dying in childbirth ?) and he came up with the idea of creating not just one Horcrux, but six, leaving a seventh soul fragment in his body. Note that by the time Voldemort has the conversation about Horcruxes with Slughorn, he has already murdered his father and paternal grandparents and has his maternal grandfather's ring in his possession.
We know that Voldemort has used the following objects as Horcruxes:
1 - Diary of 16 year old Tom Riddle which he left with Lucius Malfoy, who then planted it on Ginny Weasley just prior to her first year at Hogwarts (The Chamber of Secrets, Ch. 4) Harry destroys it by stabbing it with a fang from the Basilisk (Slytherin's monster) that was hidden in the Chamber of Secrets (CoS, Ch. 16), then returns the remains of the diary to Lucius Malfoy (CoS, Ch. 18).
2 - Marvolo Gaunt's ring which Tom Riddle hid in the ruins of the Gaunts' house. This was a large ring apparently made of gold and set with a black stone which was engraved with the Peverell coat of arms. According to Marvolo Gaunt (HBP, Ch. 10), the ring has been in the Gaunt family for centuries. After Marvolo dies in Azkaban, the ring is passed to his son Morfin and is then stolen by Marvolo's grandson, Tom Riddle, on the night that he framed Morfin for the murders of Tom Riddle Sr and his parents. Tom Jr openly wore the ring for some time but he appeared to stop wearing it once he had turned it into a Horcrux. Dumbledore found Tom had used magic to conceal the ring in the ruins of the Gaunts' house. Dumbledore destroys the Horcrux in some unexplained fashion that leaves the stone cracked down the middle. Dumbledore himself wears the ring for a while.
3 - Salazar Slytherin's locket which Tom Riddle hides in a coastal cave. This heavy gold locket carries the mark of Slytherin (an ornately serpentine S) and was inherited by Merope Gaunt (later Merope Riddle), who pawned it during her pregnancy. Borgin and Burke purchase it, then sell it to a witch named Hepzibah Smith, from whom it is subsequently stolen by Tom Riddle after he murders her. (HBP, Ch. 20). The locket is removed from its hiding place in the coastal cave by the unknown R. A. B., and another locket is left in its place with a note to Voldemort. The exact location of the real locket is unknown, although many readers suspect it is the same locket found by Harry and the others when they are cleaning the headquarters of the Order of the Phoenix (12 Grimmauld Place) during the summer before Harry's fifth year (The Order of the Phoenix, Ch. 6)
4 - Helga Hufflepuff's cup is a small, magical golden cup with two finely wrought handles, engraved with a badger (the mark of Helga Hufflepuff). It is inherited by the Smith family from Hufflepuff and according to Hepzibah Smith, the cup possesses unexplored magical powers. The cup was stolen along with the Slytherin locket after he murdered Hepzibah Smith.
Potential Horcruxes include:
1 - An unknown artefact belonging to Godric Gryffindor. The only two artefacts of Godric Gryffindor that we definitely know exist are the sword Harry used in the Chamber of Secrets to kill Slytherin's basilisk (CoS, Ch. 18) and the Sorting Hat, which is revealed to have been Gryffindor's in The Goblet of Fire (Ch. 12). Both of these items are kept in the Head's office at Hogwarts. We do not know where the sword was before Harry pulled it out of the Sorting Hat in the Chamber of Secrets, so we do not know if Voldemort had access to it in the years before he attacked the Potters. However, since Albus Dumbledore had regular access to both the sword and the Hat for several years, and since he is able to detect hidden magic, it seems rather unlikely that he would not have recognised that either one was a Horcrux.
2 - An unknown artefact of Rowena Ravenclaw. It seems likely that an artefact of Ravenclaw's would bear her mark on it - either the letter R or the Ravenclaw heraldic device, which is an eagle.
3 - Nagini, the gigantic snake which Voldemort keeps at his side. Dumbledore notes that Voldemort appears to be unusually attached to Nagini, which might mean that she is not a Horcrux since Riddle hid the other Horcruxes which we know he made.
4 - Harry himself. The only point in favour of this theory is Dumbledore's belief that on the night that he murdered Harry's parents, Voldemort was one short of the six Horcruxes he intended to make, and that he would have considered the murder of Harry, in particular, important enough to be worth creating a Horcrux. I have already stated my own argument against this theory.
The other question is where are the remaining Horcruxes hidden, if they are inanimate objects ?
It seems likely, from what we know of the Horcruxes that have been found, that the likeliest hiding places are those places which were significant to Voldemort. Therefore likely hiding places would appear to be the Muggle orphanage where Riddle was born and stayed when he was not at Hogwarts; Hogwarts itself; and the Riddle House. My own thought, on my second reading of HBP, was that the mysterious Room of Requirement in which Harry hides his Advanced Potion Making book might contain one of Voldemort's Horcruxes; it's described as being full of hundreds, possibly thousands, of magical items (HBP, Ch. 24). It would give Harry at least one reason to go back to Hogwarts - although he would need to know (as Dumbledore apparently does) how to detect the aura of a Horcrux if he was to be able to spot one in there.