I suggest reading "Words and Power" by Telenger. We know invocations or a string of words can cause magicka to manifest physically (or mentally) through runestones and the Thu'um.

"A Tragedy in Black" states that magic is equal parts thought, intent, will, and emotion.

Certain subschools of magic, like Mysticism, seem to require very intricate and out-of-the-box thought process to actually materialize and use (see "Mysticism - Unfathomable Voyage" by Tetronius Lor).

So in summary magic can be cast through a variety of ways, but in my opinion they all tie back to the aforementioned quartet of thought, intent, will, and emotion which I see as the fundaments of casting magic.

There are different ways to think about it, and that's what different schools of magic tend to do - it might be helpful to interact with the Psijics here, because the fundamentals of magicka is what they study. This is also very much what the alteration school is about. Typically, as far as where the magic comes from, I think it is best defined as a stream of power that originated with powerful entities at the beginning of creation, and was from there infused into everything. In the present, magicka has these sort of invisible threads connecting everything, and the "threads" have morphed and moved to create places where magicka flows stronger or weaker, kind of like rivers, starting from pools of magicka or other magicka generating sources. This is somewhat similar to the "force" definition from @Nestor . @jmaes is describing how mages supposedly tap into these invisible magicka rivers, and translate that power or energy into a physical manifestation. His is hardly an exclusive list, but it is a good guide to how the thinking in Tamriel goes. Some are said to be born with some way to generate magicka and they just channel it from themselves naturally (like I would imagine most monsters do), some research and use mechanical knowledge to manipulate magicka, and some simply find items that have been infused with magical properties to activate in some other way. It is generally accepted though that almost everyone has at least some capacity to tap into the magicka that runs through Tamriel.

For my part, in my personal RP, my Khajiit is actually from an island separated from Tamriel entirely, where magic study is different. In his eyes, to make a long story short, channeling lightning requires the same type of magicka channeling as manipulating time. In Tamriel, the schools of magic generally describe different ways of channeling magicka as well as their different intended purposes and uses. These two are not really mutually exclusive, because they use the same method of extracting magicka from sources and channeling it, and the definition of how to actually turn the magicka into a physical manifestation is left so open to interpretation.