Muggles' Guide to Harry Potter/Magic/Accio

Overview
Accio is the Summoning Charm; it can be used to bring an object to the spell-caster.

Extended Description
There are two instances in Harry's fourth year where he is witnessed using this spell (outside of classes and practice or homework). First, in the First Task of the Triwizard Tournament to Summon his Firebolt broomstick; and secondly, to retrieve the Triwizard Cup when dueling Lord Voldemort.

In his sixth year, Harry tries to use this spell to retrieve his wand after Draco Malfoy has put him in the Body-Bind Curse on the Hogwarts Express; he fails because he is not holding his wand. Harry later uses this spell in an attempt to Summon the Horcrux from the middle of the lake in the cave, but is intercepted. Again he uses this spell in Hogsmeade to Summon some of Madam Rosmerta's brooms, on which to fly to Hogwarts.

In his seventh year, Harry, Ron, and Hermione use this spell a number of times to bring things to them, or try to. Some objects, such as the Sword of Gryffindor, prove unresponsive to this charm; Hermione suggests that there are ways of preventing the Accio charm from working.

There are various other instances where this spell is used by other characters.

Analysis
The name of the object can follow the spell word; Harry uses that form, as "Accio Firebolt", to summon his broom. It seems that is an aid to concentration, as the spell-caster should be concentrating hard on the object being summoned. Early in Harry Potter and the Goblet of Fire, Mrs. Weasley uses the unadorned "Accio!" to relieve the twins of the prototype charmed sweets they are carrying, before they set off to the Quidditch World Cup, which shows us that naming the object being Summoned is not always necessary.

Despite its apparent simplicity, the Summoning charm is apparently fairly complex. In Harry Potter and the Goblet of Fire, we learn that Professor Flitwick has assigned several books on the subject for homework, and Harry has significant trouble with it, being unable to master it until immediately before the First Task. After the Task is completed, Professor Flitwick apparently thinks it worth discussing the Summoning charm with Harry. Flitwick apparently feels that Harry's charm was performed perfectly, which, along with Flitwick's desire to discuss it, would lead us to think that this charm is seldom performed correctly.

One has to wonder what could make this apparently simple charm such a major concern. One possibility is that the charm requires a complete awareness of what it is that you are summoning, where it is, and where you want it to end up. This would be particularly true when the item being Summoned, like Harry's broom, is out of sight. We have already determined that many spells, notably Portus, Apparition, the Patronus charm, and Riddikulus, require a significant mental exertion; so it is not unlikely that the Summoning charm, and probably the analogous Banishing charm, would also require significant mental effort. It is entirely possible that the use of the name of what one is summoning is merely an aid to concentration; that it is the mental image of what one wishes to summon that is the determinant of whether this charm works or not.

The spell's name's origin is directly from Latin. "Accio" translates directly to "summon".