Arcane Magic is what Saito uses. its essentially stuff like magic missile and things like that. More mana or arcane energy based magic along with more spells based upon practicality as well, much more 'scientific' in a sense. 

Elemental magic is more magic that focuses around a specific element and its properties. 

Magic has three stages of advancement. 

Basic, Intermediate, and Advanced. 

Basic is usually Ritual. stuff that requires materials and/or preperation time to use. For elemental magic, basic is elemental summoning (like Ren with Undine)

To learn magic you need at least an Int of 20 and Knowledge Rank 10. Cause you have to be really smart to use it. Intermediate Magic requires Knowledge rank 15 and Advanced Magic requires Knowledge Rank 20 (which is max rank)

Sasha meets the Int requirement, but she needs 4 more ranks in knowledge for the Knowledge requirement.