Magical Discipline
Not every spell can be placed neatly in categories, but this is a general breakdown of magical disciplines.
See also Spell Branch
Counter-Spells
Spells the neutralize or counter-attack other spells, both generally but more often specific spells. When counter-spells have a more general use, they could also be considered Defensive spells. When a counter-spell is a counter-attack, it can also be categorized as a jinx, hex, or curse.
Jinxes, Hexes, and Curses
Spells meant to harm others. They range from irritating (jinxes) to a bit nasty (hexes) and downright awful (curses). In everyday life the distinction between these types of spells is loosey goosey, but the Wizenagamot has much clearer guidelines. Some spells are not sinister on their face, like a Levitation spell, but can be used offensively, like dropping someone off a cliff. Those sorts of spells still aren't considered jinxes, hexes, or curses.
Defensive
Any spell whose primary purpose is self-defense. These can include counter-spells. Common defensive spells include the Sheild Charm (Protego) and the Disarming Charm (Expelliarmus).
Enchantments
Enchantments are usually spells placed on a person or object and endure over time without sustained concentration from the caster. The magical ceiling of Hogwarts castle or the Slug-Vomiting Hex are good examples of enchantments. Most Transfigurations can be considered enchantments, as can many jinxes, hexes, and curses. Many enchantments can be neutralized by the General Counter-Spell (Finite Incantatum).
Healing
Healing spells cure or treat illness and injuries, both mundane and magical. Healing spells can be either Charms or Transfigurations.