Muggles' Guide to Harry Potter/Magic/Incarcerous

Overview
Incarcerous causes ropes to appear out of the end of the caster's wand, which then bind and (usually) gag the subject of the spell.

Extended Description
When we first see this spell, it is being used by Professor Snape to bind Sirius Black and Remus Lupin in The Shrieking Shack. It is used a number of times after that, notably by Professor Umbridge, to bind the Centaur Magorian, an action that triggers a mass attack on her by the remainder of the herd.

Analysis
Branch of Magic: Incarcerous is difficult to categorize into one branch of magic due to its dual function. Most prominently, it is a transfiguration spell—conjuring to be specific. It also could be considered a charm as it is giving the ropes the ability to attack a specified target and bind to it without further magic being performed.

The Rope's Physical Properties: It would appear that the attributes of the conjured rope varies and most likely is the choice of the caster. In Harry Potter and the Prisoner of Azkaban, when Severus Snape nonverbally casts the spell, "thin, cordlike" ropes bind Sirius Black and Remus Lupin. In Harry Potter and the Order of the Phoenix, when Dolores Umbridge uses the spell, this time verbally, once more cords are produced—with no detail as to their thickness. The final time we see the spell cast in the series is in Harry Potter and the Deathly Hallows. A Death Eater casts this spell on Ron, leaving him bound tightly in thick ropes.

One possible explanation of the discrepancies in rope size is that the size could have a great deal to do with the level of power that the caster has. It is inferred throughout the final books that Umbridge was relatively weak, magically, and therefore her Incarcerous, while effective, did not produce ropes as thick as the Death Eater's. On the other hand, it is reasonably obvious that Snape is among the most powerful wizards in the story. Yet, his version of the spell, non-verbal though it may be, produces only thin cords, though these cords are quite adequate to bind both Sirius and Lupin. Quite possibly, for the relatively unsophisticated Death Eaters, bigger ropes are obviously better, while perhaps the cords that Umbridge produces are the best she can manage. Meanwhile, Snape produces just enough strength to do the job, perhaps knowing that binding with smaller cords will also hurt more if one struggles against them.

We have noted in other spells that there is a definite mental component: when creating a Portkey, for instance, one must visualize the type and destination(s) of the Portkey one is creating, while the Riddikulus and Patronus charms have the mental component explained in detail when they are first introduced. It is safe to assume that the type of binding produced, and how those bonds are to be tied, must be visualized when casting the spell.

Etymology: Incarcerous comes from the Latin word "carcerate" meaning prison or cage. Since the incantation causes ropes to be expelled from the casters wand and bound tightly around the intended target, this makes sense since it is detaining the victim, much like a prison or cage.