See DEATHLY HALLOWS.
Hand of Glory
A withered hand which, when a candle is inserted into it, gives light only to the holder (CS4).
NOTE: The Hand of Glory is drawn from English folklore, described as a withered human hand whose fingers, when lit like candles, cast a charmed sleep on members of the household so the thief can rob the house. 
Magical toy for playing Hangman; one of Weasleys' Wizard Wheezes.
Bill Weasley once had a penfriend from Brazil who was offended when Bill couldn't travel all the way to South America for a visit, so this person sent him a cursed hat that made his ears shrivel up (GF7).
The Headless Hat was invented by Fred and George Weasley. When placed upon a person's head, the head will become invisible, making it look as if the wearer is completely headless (OP24, OP26)
The Shield Hat was invented by Fred and George Weasley. It looks like an ordinary hat but functions as a Shield Charm, protecting the wearer against minor to moderate hexes and jinxes (HBP6).
The twins intended this as a joke item ("challenge your mate to jinx you while wearing it"), not knowing that many people can't perform a decent Shield Charm. The Ministry of Magic ordered 500 of them for all its support staff (HBP6)
See SORTING HAT.
Hiccough Sweet (U.S.: Hiccup Sweet)
Presumably one of these sweets makes the eater begin to hiccup.
When house points are awarded, gemstones fall from the upper bulb into the lower bulb; when house points are taken, gemstones (if there are any) retreat into the upper bulb from the lower bulb (OP28, OP38).
It is possible, though not common, for a house to be in negative points; judging from Snape's reaction when he was about to put the Gryffindor hourglass into this state, some sort of silent warning may be triggered to inform the point-taker when this is about to occur (OP38).
Gryffindor's hourglass contains rubies, Ravenclaw's sapphires, and Slytherin's emeralds (OP28, OP38), reflecting the respective house colours. Presumably Hufflepuff's hourglass contains some sort of yellow gemstone, but this is not stated.
A Howler is a nasty letter sent to tell someone off. It arrives in a red envelope, smoking slightly. It must be opened immediately or it explodes. Upon opening, the Howler screams at the recipient in a voice magically magnified for maximum effect. Students at Hogwarts occasionally receive Howlers from their parents.
After the flying Ford Anglia incident, Molly sent Ron a Howler to tell him off (CS6). (Fear of getting another Howler was what kept Ron from writing home and asking for a new wand to replace the one broken in that incident.)