The Aether is the resource used to perform magic. It's not a substance as we know it. It's got no chemical composition and it has no place in the periodic table. It simply is. In order to use the aether for magic it first needs to be channeled and then woven.
This is the process of drawing from the Aether and transforming it into a stream of magical energy that can be split and woven. The ability to channel Aether is separate from the ability to weave it and it is rare that someone is able to do both. However, it is possible to channel aether for someone else to weave and it's not uncommon that mages work in pairs where one of them is the dedicated channeler who provides the other with a steady stream of magical energy.
Channeling the aether prolongs the life span of the channeler and improves their physical health. More details about this can be read in the article about it copied from the blog here.
Weaving is the process of manipulating the channeled aether into a desired effect. At its most basic this is simply splitting the stream of channeled aether into two strands and twisting them around each other. This way many simple effects can be achieved. Simple effects are things like pushing objects around or heating them up (cooling something down is slightly more complicated).
For more complex tasks the weaver is required to split the stream into multiple strands and then weave them together in different ways (hence, weaving).
All elves are able to both channel and weave the aether, or at least they should be. Among elves being unable to wield magic is considered a handicap on par with being lame or mute.
This doesn't mean that all elves are wizards or magicians or even that they're particularly skilled users of magic. Wielding magic, like so many other things, requires both skill and talent and just being able to do something doesn't mean being able to do it well.
Anyone can buy a guitar, but there's a long step from owning a guitar to being able to make a living out of playing it. The same principle applies to elves and magic. They're able to do it, but the majority of them don't actually bother unless they have to.