Arithmancy
Arithmancy is a branch of magic that is concerned with the magical properties of numbers; someone who practices Arithmancy is called an Arithmancer. For example, in the 1200s, Bridget Wenlock, a famous Arithmancer, discovered the magical properties of the number seven. An O.W.L. in Arithmancy is required to apply for a curse-breaker's job at Gringotts.
Professor Vector was known to teach Arithmancy at Hogwarts. In her class, students were expected to write essays and to be able to understand complicated number charts, which was part of their homework. Hermione Granger appeared to be the only Gryffindor in her year that attempted an O.W.L. in this subject .
Charms
Charms are a type of magic spell, concerned with enchanting an object to behave in a way that isn't normal for that object. For example, the Summoning Charm brings an object to the caster. Charms is also something of a catch-all for spells that aren't Transfiguration (spells that change the inherent nature of an object). In some sense, if a spell isn't Transfiguration, it's probably a Charm.
A Charms spell might cause something to flash different colors. It might cause an object to levitate or even fly through the air. Charms can make a person laugh, or dance, or even create a bubble of breathable air around a person's head. In all of these cases, the object or the person doesn't really change. They just do something unexpected.
Some Charms can be extremely powerful. The Fidelius Charm, for example, can completely hide a person or a place in such a way that no one can find them unless a Secret Keeper exposes their location. Memory Charms can be so strong, that they completely remove a person's memory or even damage their mind permanently.
Charm spells are in some ways the opposite of Curses: many Charms seem to have in inherent positive tone , while Curses have an inherent negative one. This is not to say that Charms are weaker magic; a well-chosen Charm is a powerful magical tool against Curses, Jinxes, and Hexes. Professor Flitwick, the well-known Hogwarts Charms teacher, was rumored to have once been a dueling champion, after all.
The Dark Arts
The Dark Arts differ from other forms of magic, in the intent of the wizard using it. Most magic is relatively neutral; it can be used for bad or good. Some magic, however, is evil in its intention through and through. Spells of this kind are often called curses. Curses are spells that are often intended to cause harm to another person. This intention to do harm places that spell into the realm of the Dark Arts. However, simply casting a Curse spell doesn't mean that a person is using the Dark Arts. Ultimately, the deep, true intention of the caster is what makes the difference.
This is a difficult distinction to make in many cases. However, understanding the difference between acceptable and Dark magic is of key importance for witches and wizards in training. This is why Defense Against the Dark Arts is such an important class for students at Hogwarts. Some other schools have a reputation for teaching the Dark Arts, not simply Defense Against the Dark Arts. Durmstrang is such a school.
Dark Magic spells
Typical Dark Magic spells are called Curses. There are offensive spells that shoot out of a wand like a gun; curse energy causes physical damage to things it hits besides the magical effect. Hexes and jinxes are lesser spells that adversely affect the target. As mentioned above, these spells are not necessarily Dark Magic. The intention of the caster is what actually makes the difference.
Dark Magic is more than simply curses, however. Magic that involves tampering with the free will of another person (i.e., the Imperius Curse), or which kills another person (i.e., Avada Kedavra) would be considered Dark Magic indeed. Voldemort spent years in magical research into ways of becoming immortal. This pursuit is also an example of Dark Magic, presumably because it tampers with the natural order of things.
Herbology
Herbology is the study of magical plants and fungi, including their care and their magical properties and uses. Some magical plants form important ingredients in potions; others have magical effects in their own right.
At Hogwarts, all students are required to attempt an O.W.L. in Herbology. Therefore, all first through fifth year students take the class. Herbology classes are held in the greenhouses. The classes involve plenty of hands-on activities, including pruning alihotsy bushes, repotting mandrakes, and harvesting bubotuber pus.
(Info obtained from HP-Lexicon & HP Wiki)