Squib

"A Squib is someone who was born into a wizarding family but hasn’t got any magic powers. Kind of the opposite of Muggle-born wizards, but Squibs are quite unusual."

- Ron Weasley discussing Argus Filch

A Squib is a non-magical person who is born to at least one magical parent. Squibs are, in essence, the opposite of Muggle-born witches and wizards. They are rare and are looked upon with some degree of disdain by some witches and wizards, particularly pure-bloods. J.K. Rowling has stated that Muggle-born witches and wizards are descended from Squibs who married Muggles; the magical gene resurfaces after many generations unexpectedly.

Differences Between Squibs and Muggles
In addition to being born into magical rather than Muggle families, Squibs are different from Muggles in that they notice and comprehend the wizarding world. Things that are hidden from Muggles with spells — rather than by the very nature of the hidden thing, such as Dementors — seem to be visible to Squibs. Although they cannot see Dementors, Squibs are nevetheless familiar with the effects a Dementor's presence will have, as demonstrated by Arabella Figg describing one to the Ministry of Magic. They also seem capable of forming wizarding-like relationships with animals, such as Argus Filch with cat Mrs. Norris.

Attitude Toward Squibs
"You disgusting little Squib, you filthy little blood traitor!"

- Marvolo Gaunt to his daughter, whom he believed to be a Squib

The Ministry of Magic does not keep records of Squib births, a sign of wizard society's general disregard for them. The common practice with Squibs is to send them off to Muggle schools as children and encourage them to integrate with the Muggle community, which is much kinder than letting them live in the magical world where they will always be treated as second-class citizens. Some, however, choose to remain in the wizarding world regardless. Some pure-blood families, such as the Blacks, disown any Squib members and remove them from their family trees.

Even families that are tolerant of Muggles and Muggle-borns seem to regard Squibs poorly. For example, the Weasleys seem embarrassed to have a Squib who works in the Muggle world as an accountant in the family. Also, many wizarding families are anxious to see early signs of magical ability and are upset by the prospect that a member might be a Squib. Neville Longbottom and Merope Gaunt were both thought to be Squibs by their relatives because of this, though ultimately both of them were magical.

Known Squibs

 * Marius Black
 * Arabella Figg
 * Argus Filch
 * The Thurkell brothers

Believed Squibs

 * Neville Longbottom — discovered to not be a Squib when his Uncle Algie accidentally dropped him out of a window and he bounced down the road.
 * Merope Gaunt — discovered not to be a Squib when she married Tom Riddle Sr. after running away from home. She was able to develop her magical abilities freely, and had not displayed them before due to fear of her father and brother.
 * Ariana Dumbledore — unable to control her magic after being traumatized by an attack by Muggle boys, and kept hidden by her family.
 * Unnamed second cousin of Molly Weasley (assumed)

Behind the scenes

 * The term likely comes from the English expression "a damp squib" (dud firecracker), an expected delight that disappoints.

Notes and references
Cracmol