There are a few rules around magic in this story. 

Natural Magic vs Spells

The differences here are based on where the source of the magic comes from, not the result. A common example in the story is the magic of shape-shifting (natual magic) and Glamour (a spell). Both will turn you into a different shape, but natural magic means that the creature itself has the ability to will themselves into that new shape. The spell requires an object or ritual to achieve the same result. 

Common Types of Magic

Elemental Magic - control of the elements of fire, wind, water and earth. 

Travel-Based - get a body or object from here to there in no time at all. Portals, transport spells, and plane jumping are examples. 

Protective Spells - Wards, Sigils, Runes and anything that prevents unwanted magic or any magic to be performed. These can be set to a location or person. 

Glamour - a spell to change the appearance of a body. A person can be glamoured to look like someone else although that is a discouraged practice.

Shape-shifting - ability to change the appearance of a body. Oddly enough, shape-shifters cannot shift themselves to look like someone else. 

Dream Magic/Mind Magic - perceiving or communicating in someone's subconscious. Differences whether the body is awake or asleep. Dreams can also be prophetic visions from the future. 

Communication Magic - the ability to talk or send a message over a great distance. Like email. 

Trap Spells - spells to immobilize another person when triggered. "Trance" or paralysis is common.

Reincarnation - natural reincarnation occurs after death, the creature comes back to life into the prime of youth again. Powerful creatures like djinn or eastern dragons can help other creatures reincarnate. 

Ritual Spells - these often require specific ingredients, conditions and such to perform a task. Reincarnation can be done as a ritual spell for spell inclined creatures. This reincarnation is a proper rebirth so the reborn creature will not remember their previous life. (Of course they are often able to remember parts of it through other spells.)

Scrying - it's like searching the internet for a solution to your problem. 

Enchantment - attaching a spell to an object. For example, Kichiro's glasses help him read faster (He actually has perfectly normal vision and doesn't need glasses.) and Katsuro has an amulet to help him shape-shift.