Background

A Horcrux is the word used for an object in which a person has concealed part of their soul.

We know it is seriously evil - not even dark magic books talk about it.

“I haven’t found one single explanation of what Horcruxes do!” she told him. “Not a single one!

I’ve been right through the restricted section and even in the most horrible books, where they tell you how to brew the most gruesome potions — nothing! All I could find was this, in the introduciion to Magick Moste Evile — listen — ‘Of the Horcrux, wickedest of magical inventions, we shall not speak nor give direction’…I mean, why mention it, then?”

A spell is used:

“There is a spell, do not ask me, I don’t know!” said Slughorn

It involves ripping the soul apart:

‘And the more I’ve read about them,’ said Hermione, ‘the more horrible they seem, and the less I can believe that he actually made six. It warns in this book how unstable you make the rest of your soul by ripping it, and that’s just by making one Horcrux!’

We know that you can create several, though Voldemort was the first to do this:

“You heard Voldemort, what he particularly wanted from Horace was an opinion on what would happen to the wizard who created more than one Horcrux, what would happen to the wizard so determined to evade death that he would be prepared to murder many times, rip his soul repeatedly, so as to store it in many, separately concealed Horcrux. No book would have given him that information. As far as I know — as far, I am sure, as Voldemort knew — no wizard had ever done more than tear his soul in two.”

We know you can accidentally create semi-horcruxes (possibly after you have made one before):

‘You were the seventh Horcrux, Harry, the Horcrux he never meant to make. He had rendered his soul so unstable that it broke apart when he committed those acts of unspeakable evil, the murder of your parents, the attempted killing of a child. But what escaped from that room was even less than he knew. He left more than his body behind. He left part of himself latched to you, the would-be victim who had survived.

Requirements

We know that murder must be committed:

“How do you split your soul?”

“Well,” said Slughorn uncomfortably, “you must understand that the soul is supposed to remain intact and whole. Splitting it is an act of violation, it is against nature.”

“But how do you do it?”

“By an act of evil — the supreme act of evil. By commiting murder. Killing rips the soul apart. The wizard intent upon creating a Horcrux would use the damage to his advantage: He would encase the torn portion —”

We can deduce that the object being turned into a horcrux needs to be near or in the possession of the person making the horcrux. Otherwise, Voldemort wouldn't have gone to all the bother of stealing them from Hepzibah Smith. However, details are hazy about whether you need to be present yourself (we don't know where Voldemort was when he made the cup horcrux).

We also know that the spell is possible to perform in a weakened state, and it is not obvious to observers that a horcrux is being made. Wormtail was present when he killed Frank.

Incantation

We don't ever hear the actual spell used to make one. I am not aware of any quote from JKR about this.

We know, however, that avada kedavra works for the first part - the murder. It is also possible via a proxy murder, which Voldemort commits through a snake, and also possibly house-elf:

After an interval of some years, however, he used Nagini to kill an old Muggle man, and it might then have occurred to him to turn her into his last Horcrux.