When a spell is cast, you will most likely create a GameObject that represents the spell's area of effect. This object can move, will probably have a collider, as well as one or more visual components (particles, mesh, etc.) Maybe the audio for the spell will also be on it- you'll hear the sound position change in 3D that way.
You'll probably set up one or more spell objects in this way and store them as prefabs, to be instantiated later whenever a spell is cast.
You'll also most likely have your own script(s) attached to this GameObject. These scripts will make it fly across the scene, shrink or grow over time, play sounds, etc.
Given this setup, here are two ways to do what you want.
1) Let's say you have a script called Spell that you've dragged onto this GameObject. Any public variables of Spell will appear in the inspector. So you could have a variable "spellType" that represents what type of spell it is. Or, "fireDamage", "waterDamage" could be integers of how much of each type of damage is done.
If the Spell code is what detects the collision, which I think is a good way to do it, then Spell could SendMessage to the objects it collides with. Those objects, if damageable, would respond to the message, adjusting their hit points and so on.
2) If you spells are fairly similar in appearance and behavior, then method 1 may be good enough. Otherwise, it's a good starting point and we can take it to the next level. Instead of one general purpose Spell class with variables that control the effects, you could make vastly different scripts and gameobjects. Maybe one spell is more like a projectile, another is a wall, and a third is a sleep spell, for example. Each would have its own class (FireSpell, WallSpell, etc.) and they could do vastly different things. One might stay put, another might not be visible at all, but instead would cause an enemy to move slowly for a while.
So to summarize, spell scripts send messages to things the spell hits, and the enemy scripts can respond or not, depending on what type of damage they take, whether they get sleepy, etc.