I haven't tried all the specializations in Inquisition yet, but I'll give you the rundown as best as I can:
Knight-Enchanter is a mainly defensive specialization. You have a passive that generates barrier when you do damage, which makes you almost invincible as long as you keep attacking. For the most part you engage enemies at melee range and continuously whack them with Spirit Blade. If your Barrier does run out at any point you have an ability called Fade Cloak which makes you invincible for a short while and allows you to recover. It's not the most damage-focused specialization, but you're almost unkillable and you get to be more in the thick of things, rather then shooting projectiles from the backline.
Necromancer focuses on debilitating magic, it's sort of a mixture of Spirit magic and Entropy from Origins and DA2, if you've played those. It has some damage-over-time spells and some useful passives. I haven't tried it myself, but from what I've experienced with Dorian, it's rather underwhelming.
Rift mage is pretty fun, it adds some more offensive capabilities, some crowd control and basically all the spells weaken enemy. It's good, but it won't fundamentally change the playstyle of your mage the way Knight-Enchanter does.
Champions are the ultimate tanks. You have abilities that generate guard, make you straight up invincible and make enemies attack you. Similar to the Knight-Enchanter the Champion is basically unkillable.
Reavers trade health for damage, that's like their entire thing. The less health you have, the more damage you do. In addition they have some limited self-healing abilities. If you choose this one you should get all the normal guard-generating abilities you can. That allows you to have very little health (i.e. a lot of damage) but you don't necessarily die immediately.
Templar is sort of a support specialization. It's meant to counter enemy mages, demons and buff your party. Haven't tried it myself, but I hear it's pretty good.