Muggles' Guide to Harry Potter/Magic/Undetectable Extension Charm

Overview
The Undetectable Extension Charm allows a bag or other container to be larger on the inside than it is on the outside.

Extended Description
While this charm is mentioned by name only once, it is used in many places within the series, and extensively through the seventh book. Mr. Weasley has used this charm on his flying car, which we see in Harry Potter and the Chamber of Secrets, to make the inside of the car larger than the outside. The tents that Mr. Weasley borrowed for the Quidditch World Cup make use of the same charm, to be larger inside than out, as do, we are led to believe, many of the other tents there. At Bill and Fleur's wedding, Hermione drops her small, beaded bag, which falls with a suspiciously heavy thump. Later, in the Tottenham Court Road, Hermione produces Muggles clothes and the Invisibility Cloak out of the bag. When she shakes the bag as if to demonstrate, we hear some large things inside it falling over; Hermione, dismayed, says that will be the books, which she had carefully stacked. We later see her forcing the large portrait of Phineas Nigellus into it, and still later she will produce one of the tents that Mr. Weasley had borrowed for the Quidditch World Cup out of the same bag. As these tents use the same extension charm, clearly one charmed object can be safely placed inside another. Almost all of the needs of the group, as they wander England, will be met by the contents of this one, small, beaded bag. When Harry, Ron, and Hermione are captured, Hermione tucks the beaded bag into her sock, where it remains undetected.

Analysis
Given how much is actually stored in the bag, a tent, a large portrait, a small library of textbooks, and clothing and supplies for a party of three, the bag should be far too heavy to lift; and in fact, at one point when it is dropped, it does make a suspiciously heavy thump.

It is noted in an earlier book, Harry Potter and the Chamber of Secrets, that Mr. Weasley's flying Ford Anglia has been charmed so that the inside is larger than the outside. This is clearly either the same charm, or else is one that is closely related. The fact that Mrs. Weasley believes that this is a feature of Muggle design would indicate that this is an unusual spell, or that Mrs. Weasley is being sarcastic. No mention of a sarcastic tone of voice is made, though, so we have to assume that this is advanced enough magic that she does not expect it.

It is also mentioned that the Ministry cars that take Harry, Hermione, and the Weasleys from The Leaky Cauldron to King's Cross Station in Harry Potter and the Prisoner of Azkaban are similarly charmed so their interiors are larger than their exteriors.

Further evidence that this charm is advanced comes from the fact that it is apparently never used on Hogwarts school trunks. It would certainly be a useful thing to be able to store all of your year's supplies for school in a robe pocket, but we never see any students doing so.

Greater Picture
As a much more gruesome example of the use of this particular spell, consider the case of Bathilda Bagshot in Harry Potter and the Deathly Hallows. We have been told on at least two occasions that Nagini has been offered a human for "dinner". When Harry is tied to the gravestone prior to the battle in the Cemetery, Voldemort promises Harry to Nagini. Later, Voldemort murders Charity Burbage, and then summons Nagini to dinner.

From these two instances, we can clearly see that Nagini must be significantly larger than a human being. Yet, shortly after Harry meets Bathilda in Godric's Hollow, Nagini emerges from Bathilda's neck. Some variation of the Undetectable Extension Charm must have been used to allow all of that snake to fit inside Bathilda's animated body.