Magic a stored potential to use human will to alter the world in some way. Whether that be superheating air till it spontaneously combusts and flings itself at someone in the form of a fireball, the levitation of an object or the fiddling with another persons mind to compel them to act in ways other than they would usually. The bigger the action the more magic required.
Potential which is stored is burned off when a person's will is used to effect change. When magic has been worked a residual amount of magic, a constant small percentage of the total magic used remains behind, irradiating the area, person or object it was used on.
Magic originates from those with magical talent.
Imagine a cup, which can hold magic like it were water. Mages are born with the natural ability to refill their cup, the size of their cup (their total potential) and the speed at which is refills (their ability to regenerate magical potential) starts off low and slow, over time they are able to work more and more impressive effects as their total magical capacity increases. With each refilling of the cup, the regeneration gets quicker and the cup gets larger.
This is the reason for the 10 year lag between a person first displaying talent at roughly age 8 (the age at which a persons cup grows large enough to store enough magic to make noticeable effects but with a very slow regeneration rate) and their becoming a fully fledged mage (this is roughly the amount of time it takes to build a large enough reserve and regen time for making magical workings practical.) It is a self-reinforcing almost exponential growth after a certain point.
All people, even those who do not produce magic have 'cups' however because they do not refill by themselves they fill from background magic if at all meaning that if they grow they do so incredibly slowly.
After the magic pipe infrastructure was put in place, most of the population was put within much closer regular proximity to constant magical potential and working (the pipes and the arcane-engines) meaning that their 'cups' were being refilled more often, allowing the growth of their ability to retain magic. For the mundane person this growth of magical ability is linear however due to their reliance on background magic.
The Knights explained this sudden growth of low level ability by spreading the story that Mages used to steal magic from everyone. The opposite is true, Mages are the only source of magic.
The machines were designed during the local Mage wars so that the power of hostile Mages could be put to use after their capture. During the main war those within the Tower used it to canabalise their own numbers for extra juice.
Hope that clarified more than confused. I wanted the magic system of the world to be confused and lied about amongst the characters not those writing

Also I haven't given the city or the setting a name yet, any suggestions? I suck at naming stuff.