Bending is the only power in Nautilus. (While people have gotten canon abilities back, in truth that is only Bending manifesting itself as a power.) It is the act of willing things into or out of existence, whether they be objects or desires.
Bending is how all the universes were created, how Nautilus was created, et cetera. Bending is the reason a Wakened even has a body. It is also the reason why those can't die unless they Fade. Basically, if a Wakened's physical body is destroyed, they subconsciously and automatically Bend themselves a new body elsewhere in the city.
Conversely, it is not required for a Wakened to learn how to Bend. If they do not and refuse to believe in having powers, they won't have any powers.
Scientifically wise: Think of it as though the entire universe exists on a mental playing field - the world is a dream. There's only the interaction between minds and thoughts. Science need not apply, and the only limits are what the imagination can handle. Physics, energy, matter, gravity; these things don't really exist - they exist because it's convenient for them to, because we believe them to exist.
A Wakened actually have none of their original powers in Nautilus. Due to the nature of the city, reality Bends according to the wills and whims of those within it – thus, if a Wakened with powers works on re-gaining their powers, reality will Bend to their wishes and their powers will return.
However, this does not just apply to Wakened with canonical powers. Anyone can learn to Bend if they strongly desire something reality will re-shape itself. Those with powers are not limited to their canonical abilities.
Bending is very difficult and requires both a strong force of will and extreme concentration. While in theory anything can be accomplished with it, in practice it is limited by the mental strength and endurance of one and their imagination.
Imagine a world full of darkness. It's just you and a flashlight. You have no idea what kind of world you're in — if it's made of stone, if there's an ocean in front of you, if there are trees. Absolutely nothing.
The darkness is Chaos. Every possibility exists simultaneously — absolutely anything could be out there in the darkness, and you would have no way of knowing for sure without using your flashlight.
The flashlight is the Light. Light brings order to Chaos, solidifies possibilities and makes them real. Shining the flashlight onto a rock makes that rock a certainty in the darkness. It takes it out of the realm of possibility and into the realm of Things That Definitely Exist.
This is the basic idea of Bending: taking something out of all possibilities and making it real. It requires both Chaos and the Light to be possible — you need Chaos to facilitate the possibilities, and you need the Light to focus those possibilities into something concrete. The entire universe resides in this shadowy balance of Chaos and Light. As has been evidenced within Nautilus time and again, upsetting the balance can have disastrous consequences.
But how does willpower factor in? Think about it in the terms of the metaphor again. You could spin in circles and wave your flashlight around, but that's not going to help you discern your surroundings. Willpower is shining the flashlight on one spot and keeping it there until you can safely say, "Yes, that is a rock."
Let's say Person A wants to Bend a giant black spike in the Plaza of Waterfalls, and Person B doesn't want that. The two oppose then, with B trying to keep A's Bend from happening.
Basically, this is a contest of will, with the winning Bender getting what he/she wants. We expect the players to resolve these the same way you resolve any combat – logically and calmly.
Bending leaves traces of whoever did it. Everyone has a signature style, and if you're careful you can tell that say, Lelouch does his this way and Klonoa does his that way.
Angelii and Deva (but, interestingly, NOT Ashura), however, have the ability to see these traces as actual glowy symbols somewhat along the lines of Instant Runes. In truth this is just being very, very damn observant and peeling back the layers of reality to see the "programming" that lies underneath. (Though it should be noted that Angelii cannot see bending traces over the network. It is an in-person ability only.)
"So why the hell do Angelii see them and Ashura don't?!"
If Ashura are the programmers of the 'verse, Angelii are the debuggers. They have to work together to fix problems - it's part of the whole deal. Ashura may be more capable at actually dealing with Bending, but Angelii are better at spotting it, and spotting errors in their code.
That's what this should be thought of as—a programming language for the universe. Run with that as you see fit (if somebody has messy or unstable Bending, an Angelii might [after training] be able to go "you've got a hole here mang").
Each character leaves a fairly unique pattern behind around objects they bend. The patterns contain clues to the form, function, and purpose of whatever's been created - be it a glass of water or a breeze scheduled to blow at this location at 5:30 pm or a warding designed to keep all non Cybertronians out of a certain area.
In addition to Bending patters, there are also Threads, which are also colors/patterns. Only Deva can see threads, and threads represent information about a character. This is how a Deva seems to already know everyone's name and bits of their history (and they see all possible relationships/alternate universe versions of a person too).
Archive of Bending traces can be found here.
Outside of Nautilus, Bending doesn't work. Binding is basically weaving an ability or personal talent into a Wakened's own being. Since it becomes attached to the Wakened, learned abilities and powers become integrated.
Binding leaves behind a distinct Bending Trace behind for the Angelii. With this, they have an idea of what the ability or talent is and how it is used.
One should note that it takes a great amount of Will to sustain these abilities. Without practice, a Wakened can exhaust themselves. Note: It would be wise to use this your discretion. Choose your abilities accordingly and don't choose too many or else it may overwhelm you.
Binding can be taught to anyone by anyone who already knows how to Bind. Simple as that.