Fade is the realm of dreams and most mages draw power from the bits of Veil or Fade that sticks to them and perform their magic. Some (Rift Mages) can draw directly from the Fade and be exceptionally powerful as a result of it. Other use Fade creatures, pushing them into bodies or to terrify someone.
Blood can be the Blood in the conventional sense as well as Lyrium, which is the blood of the Titan. A mage draws from the power of the Blood or Lyrium to cast their spells instead of the Fade. This is why Solas mentions using Blood magic weakens a mage's connection to the Fade. It is because you are drawing from another source. Think of World of Warcraft, where there is Arcane magic and Fel magic. That's the difference between Blood magic and Fade magic. As for Lyrium magic, it does not work in the conventional sense in that it seems to function as a counter to Fade magic. By this I mean that Lyrium derived abilities such as the ones used by Templars, are abilities that reinforce reality and block the Fade from being used. In my opinion, the relationship between Fade magic and Lyrium magic similar to the relationship between Yin and Yang. They are harmonious opposites that cancel each other out. That is why Templars are able to cancel out effects of Fade magics and Fade spellcasting. Interestingly, I think this is why Templar abilities do not work on Blood mages. This is because Blood magic is not Fade magic and therefore cannot be countered by Lyrium magic. What's more, Lyrium magic works very much like how a Cleric or a Paladin in Dungeons and Dragons use their spells, instead of Wizards or Sorcerer. Lyrium is the blood of Titan, a godlike creature and when you use the power of its blood, you are essentially tapping into its "Divine" power. This is no different than a Cleric tapping into the power of Pelor's divinity and faith. Fascinatingly, this would make Templars be Blood Mages. I would love to the see the faces of Andrastians when they learn of this truth.
Taint is the curse or affliction carried by Darkspawn and by other Blighted Creatures. It is its own source of magic because Darkspawn do not use the Fade to cast their spells and, according to Avernus, Fade creatures become unnerved when one uses the Taint to cast spells and the Taint is powerful in its own way. Simply put, the magical effects and abilities that is used by someone who draws from the Taint is different from anything we have seen. Interestingly, if we look at the novel titled 'The Calling', Taint magic can be taught and learned by those who do not possess the Taint, as we see First Enchanter Remille using it and he is probably taught be The Architect. Corypheus uses this as a source of magic as well, albeit via the Red Lyrium and Fen'Harel's Orb which amplifies his power. As for where the Taint comes from, I will hazard a guess and say that Taint comes from the Void and that its power is used by the Forgotten Ones, who are another faction of Elves that were portrayed as the gods of pestilence, terror, malice and spite. Andruil also managed to gain some power over its usage by forging armors and weapons out of it.
Each one of these sources of magic are different and each one has certain spells that are unique to it which cannot be cast via other methods. For example, I doubt Blood Mages are capable of casting Rift Magics. I doubt normal mages are capable of mind control. I doubt normal mages and Blood Mages can block magics and manipulate the earth like Lyrium Mages can. Lastly, I doubt any normal mage, Blood Mage and Lyrium Mage can affect the Darkspawn, control demons and corrupt everything like Taint Mages can.