Ignatius Weasley, Percy: Etym: Ignatius Loyola was the founder of the Jesuits.
Igor Karkaroff: Etym: From the Viking name Ingvar(r), meaning "hero". The name of a couple Russian princes; one was the founder of the Kievan dynasty of Rus, and the other is the Prince Igor of the opera.
Imago, Inigo (OotP ch. 12): The author of The Dream Oracle.
Etym: In psychoanalysis, a subjective image formed in the subconscious that influences one's behavior without conscious knowledge.
Impedimenta: The incantation for the Impediment Curse.
Etym: Latin, "impediment".
Impediment Curse: A spell that slows or completely stops the movement of a living being. Incantation: Impedimenta.
Impediment Jinx (OotP ch. 19): Probably the same as the Impediment Curse.
Imperio: The incantation for the Imperius Curse.
Etym: Latin, "I command".
Imperius Curse: One of the Unforgivable Curses, a spell which makes the target do the caster's bidding against their will, although a strong wizard can fight the curse to some extent. Incantation: Imperio.
Imperturbable Charm (OotP ch. 4): Wards off anything from making contact with the target.
Impervius: An incantation that makes a surface repel water.
Etym: Altered spelling of impervious.
Important Modern Magical Discoveries: A book in the Hogwarts library.
Improper Use of Magic Office: The arm of the Ministry of Magic which, among other things, keeps tabs on any students who try to use magic outside of school.
Inanimatus Conjurus (OotP ch. 14): A spell to conjure up a non-living thing.
Etym: From Latin in- "not" + animatus "life, animation" and conjuro, the root for "conjuration".
incantations: The verbal components of spells. Incantations come in a variety of languages. Oddly, modern-language spells which contain verbs are in the imperative (the command form) but the Latin ones are in the indicative (that is, descriptive of action). There may be a deep reason for this, but it's probably just because Rowling was never forced to take Latin. I was, and by golly I'm going to get some use out of it. Imperative forms are provided for the curious.
Incarcerous (OotP ch. 33): An incantation that conjures ropes to bind the target.
Etym: The Latin root is carcer "prison", as in incarcerate.
Incendio: An incantation to start a fire.
Etym: Latin, "I set fire to". Imperative: incendere.
Inigo Imago: Etym: Unknown, a cognate to Ignatius, but I think it was just picked for assonance.
Inquisitorial Squad (OotP ch. 28): Umbridge's own little junior Gestapo.
Inner Eye: A term for clairvoyant ability.
InterCity 125: An engine widely used on Britain's commuter railways.
Inter-House Championship: The competition between the Hogwarts houses to see which can accumulate the most points by the end of the year.
Intermediate Transfiguration: The textbook for the third-year Transfiguration class.
International Association of Quidditch: The organization which sanctions the Quidditch World Cup, akin to FIFA.
International Ban on Dueling: The exact nature of the ban has not been explained, but since Lockhart was allowed to have a go at starting a dueling club, it's either very recent, or not a total ban, or the UK is not a signatory to it. We know for certain that Transylvania is not.
International Confederation of Warlocks: Some sort of multinational professional association.
International Confederation of Wizards (OotP ch. 7): Another variation on the International Confederation of Warlocks.
International Federation of Warlocks: Probably a typo for the International Confederation of Warlocks, or vice versa.
International Magical Office of Law (OotP ch. 7): A division of the Ministry of Magic.
International Magical Trading Standards Body (OotP ch. 7): Another division of the Ministry of Magic.
Invigoration Draught (OotP ch. 28): Something Harry had to make in Potions class.
Invisible Book of Invisibility: The most problematic book ever to blight the stockroom of Flourish and Blotts until the arrival of The Monster Book of Monsters.
Invisibility Booster: Something that can make a flying car invisible, at least to Muggles.
invisibility cloak: An extremely rare magical item which makes the wearer(s) totally invisible to normal sight, though it is penetrable with any kind of magical sight. Harry's father left one to him.
Io (OotP ch. 14): One of the Galilean moons of Jupiter.
Ivanova (GoF ch. 8): A Chaser on the Bulgarian national Quidditch team.
Etym: Patronymic from Ivan, another form of John.