"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)
Vanishing Magic consists of various types of spells and items that 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