"It was your head, Potter. Floating in midair. ... What would your head have been doing in Hogsmeade, Potter? ...Your head is not allowed in Hogsmeade.No part of your body has permission to be in Hogsmeade"
-- Snape and Harry (PA14)
Various types of spells and items make things vanish, but in different ways.
References from the canon
The basic form of Vanishing is the Vanishing Spell. Taught in fifth year Transfiguration (OP13), this spell is usually cast using the incantation Evanesco which is Latin for "vanish." McGonagall starts students out with Vanishing invertebrates -- snails --and then works them up to mice because they're more difficult (OP15). The Evanesco spell apparently doesn't need an object word (you don't have to say "Evaesco Potion," for example). The intent of the caster provides the needed focus on a particular object.(CS11)
What happens to Vanished objects?
McGonagall states that Vanished objects go "[i]nto non-being, which is to say, everything" (DH30). This does seem to fit for most circumstances, but there are a few twists. For example, a mouse made to Vanish incompletely will then exist simply as a wriggling tail. This suggests a connection to the concept of Splinching, where a person who incorrectly Apparates leaves some body parts behind. (HBP18) Splinching, though serious, is not fatal, so the body parts are presumably still connected in some way to the Splinched person. In this way, the rest of the mouse is happily existing in some other location, and the tail, while separated by distance, is still linked to the rest of the animal. It seems, then, that anything Vanished does not cease to exist. The potions both Snape and Hermione made Vanish might have ended up in a drain or other disposal location then. (OP12,OP29) However, in some cases the items made to Vanish with an Evanesco spell certainly would not be lost forever. The maps and charts that were Vanished in the kitchen of Grimmauld Place also must have moved to a cupboard or other storage place (OP5).
Other forms of Vanishing magic
A Vanishing Cabinet makes things Vanish, sending them to another Vanishing Cabinet to which it is magically connected. There is a Vanishing Cabinet on the First Floor of Howarts which is linked to another such Cabinet at Borgin and Burkes in Knockturn Alley. The Cabinet was malfunctioning for years, and people who were sent through found themselves trapped. When Montague was stuffed into the Cabinet, he reappeared some days later, crammed into a toilet. He was out of his mind in the Hospital Wing for a long time after that, recuperating. Draco Malfoy spent a considerable amount of time and effort repairing the Cabinet and used it to bring Death Eaters into the school (HBP27). It is unclear whether the Vanishing Spell is actually being used by the Cabinet.
Some magical creatures have natural Vanishing ability. The Diricawl is an excellent example of this. The phoenix can also disappear from one location and reappear in another.(FB)
House-elves have the ability to Vanish at will. This is not Apparating, although it resembles it; it is a natural ability of the elf. (CS2)
This spell effect is different in that the Invisible person or object is still very much present, just impossible for most beings to see. The Invisibility Cloak is an example of this effect. Invisible Ink is also still on the paper, just impossible to see. Invisibility, rather than Vanishing, is the spell Fred and George put on their Headless Hats.(OP24,OP26)
Another form of Invisibility is the Disillusionment Charm. This spell doesn't actually make a person invisible, but it makes them extremely difficult to see by making them blend in with whatever they're standing in front of.(OP3)
Some magical creatures have the ability to make themselves invisible. The hair, hides, or other parts of these animals are used for various Wizarding forms of Invisibilty. Examples of these animals include the Tebo, the phoenix (whose ability is more closely related to Vanishing), and the Demiguise.(FB)
Dumbledore stated that he doesn't need a Cloak to become Invisible, which could suggest that he is capable of casting an Invisibility Spell on himself. This must be rare, since no one else does it.(PS12) Harry assumed that Dumbledore could "perform a Disillusionment Charm so powerful that he made himself completely invisible without one” (DH22).
Ways to Un-Vanish Something
The Revealer is an eraser-like device which makes invisible ink appear. Moody's magical eye could see through Invisibility Cloaks. (GF25) It would seem that Voldemort in the form of a snake could as well, since Harry could see Arthur before the "cloak fell away" when he was seeing through the snake's eyes. (OP21) Anyone placed in a Vanishing Cabinet reappears after a while, although this can be a matter of several days.