Pirate questions are often difficult, because pirates as a group have been so heavily fictionalized -- and were often fictionalized even in the "golden age" of piracy (or shortly thereafter) -- that it becomes hard to separate reality from fiction. Indeed, there is evidence that later sailing crews, including pirates, took on behaviors that they may have read or heard that "pirates" did -- even if these accounts were fictional, meaning that fictional accounts of pirate activities actually became reality due to people acting out what they'd seen in fiction.
How Did it Work?
Let's answer the second part of your question first. Walking the plank is among the many brutal punishments that are associated with pirate lore, and it hits two of the major stereotypes that we think of when we think of Age of Sail pirates: it was particularly cruel to the victim, and particularly amusing for the sadistic pirates to watch. At least, that's the theory.
Victims were bound at the ankles and wrists, and potentially gagged, so that they could not swim or tread water. Sometimes they would have weights shoved in their pockets or tied to them, as well. They would then be forced to walk (or hop/sidle) a plank -- extended from the ship's side for this specific purpose -- usually by being prodded by sharp objects such as spears. Eventually, they would run out of plank or slip, falling into the water and, being unable to swim, drown. Then a hearty laugh would be had by the entire crew of the pirate ship, who are all heartless, grog-swilling men of low moral fiber.
Did it really occur?
We have some evidence to suggest that it did, rarely -- and not until after the so-called golden age of piracy was over (or at least coming to an end), perhaps long after. Perhaps the "best" (and most oft-cited) primary source for this is the confession of prisoner George Wood, who was not a pirate, but a mutineer from the merchant marine. He stated to his chaplain in prison that he and other mutineers had compelled their officers to walk the plank to their deaths (Botting, p. 58). While there is an official record of this confession from Newgate Prison, this confession is so often cited as the main "proof" of the practice of walking the plank occurring that it becomes counterproductive. It was also given in 1769, long after the golden age of piracy had ended.
There are other cited instances of plank walking that are of a somewhat dubious nature. The Times (London) reported in 1829 that a packet ship of 10 guns called HMS Redpole was taken by pirates outside of Rio de Janeiro and that the captain was put to the sword and the rest of the crew made to walk the plank... but this strikes me as journalistic sensationalism. (The Times, Feb. 1829) There is almost no way The Times reporters could have independently verified these facts.
There are a few other reported instances, usually from the early 19th Century, long after the era of the Blackbeards and the Calico Jacks. To the extent these did occur, such as the taking of a certain Dutch ship and the officers being made to walk a plank with cannonballs tied to their ankles, it seems likely that the pirates were inspired by tall tales of other pirates allegedly doing the practice, rather than the regularity of the practice itself.
A far grimmer, and more real practice, however, was not done by pirates, but slavers. While they would not often waste time with the ceremony of walking the plank, slavers, such as those of the Tecora (which was depicted in the film Amistad) would often force slaves overboard dozens at a time (generally by chaining or tying them all to each other, then chaining a cannonball or heavy stone to the final slave and pushing him over the side). In theory, slavers did this to combat famine, disease, thirst, or other problematic transport conditions. In reality, maybe they were just cruel. While this doesn't exactly go to your question (doesn't involve pirates or plank-walking directly), it does show that there were unscrupulous and cruel people willing to kill others at sea by a similar means. (Rediker)
Keelhauling
A final aside: a much more common and verified practice from the time of "walking the plank" is keelhauling (from the Dutch "kielhalen"). This was well documented and officially approved by the Royal and Dutch navies (among others), and was performed by pirates whose sailors were often castoffs from those navies. Here is a painting by Dutch artist Lieve Verschuier depicting the preparation for a ship's surgeon to be keelhauled after he failed to save the life of an admiral. This practice was far and away sufficiently sadistic and lethal enough to stand in for walking the plank, and while often reserved for "special" punishments, would not have required a ship to heave to or otherwise cease all-important movement in order to carry out. And, just to drive the point home: since these guys already had a sufficiently "tough on crime" and lurid punishment with which they were familiar (and plenty of other, less lurid ones, like simple hangings), there probably wasn't a need to dream up or implement punishments like walking the plank.
Conclusion
How did it work? For the "pirate" version: People were bound (and potentially blindfolded), and prodded with sharp objects until they walked off the edge of a plank or ladder that had been extended off a ship's gunwales or rails for this specific purpose. They would then be left to drown, being unable to swim for having been bound. For the much more realistic, "slave" version: people were tied together, then tied to something heavy, then that heavy thing was pushed overboard.
Did it really occur? For pirates -- maybe, but probably not until long after the stereotypical flintlocks-and-cutlass pirates were gone, and definitely never on any wide scale. For slaves, yes, although whether this constitutes "walking the plank" is probably up for debate.
Sources
One shit-post more, Mr. Hands, and I'll ban your head off!
Mateys,
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