I am omitting any elaboration on the true meaning of the Many-Faced God or the Faceless Men. It's intentionally kept vague, there are many fan theories.
My answer only focuses on what "no one" means, and how it is important for Arya as a Faceless recruit.
The Faceless Men have no identity
This is difficult to grasp for us humans. Everything we do is tied to who we are. So instead of considering humans, consider ants.
Ants have no concept of self-preservation. They will gladly sacrifice themselves for the colony. Everything they do, they do for the colony (which is represented by their queen).
The Faceless Men are similar. They have no identity, so as to prevent them from having selfish desires. Everything they do, they do for their ideology (which is represented by the Many-Faced God).
The face we know as Jaqen H'gar explicitly shows this to Arya, when he willingly kills himself in front of her. The person standing before Arya does not care about his body's biological survival. It is irrelevant.
"Jaqen" is nothing more than a face. Any Faceless Man can wear this face. It is not unique.
The only reason why we know "a man" by his name, is because "Jaqen H'gar" is the name he gave to Arya when he was pretending to be human. "A man" doesn't have a name, but it would be impossible for "a man" to blend in with regular humans if "a man" has no name and refuses to give one when asked.
The proof is in the pudding.
Was the previous paragraph understandable? I only spoke of "a man", I didn't specify this man's identity, I only said which false name he gave. Because his identity doesn't matter. Therefore, we forgo the need to specify an exact example, and simply speak of "a man".
This is basically what "a man" constantly does, he refers to himself as "a man", because that is all he is.
To quote Hugh Grant:
"I'm just a boy, standing in front a girl, asking her to love him."
If you remember the context of this quote, you'll know that Hugh Grant was saying that "it doesn't matter who you are, all that matters is that I am someone who loves you".
This is a bit more obvious for the waif. She doesn't have a name. If you watch the show by yourself, there is no need to ever know a name for this face.
However, she is referred to as "the waif" so that we can refer to her. This name is given to her only in the book narration/script/screenplay, and in discussions about Game of Thrones.
In-universe, Jaqen and the waif themselves never use this name (as far as I'm aware).
Jaqen speaks of "a man" and "a girl", because the Faceless Men have no specific identity. When Jaqen says "Arya Stark", he means "the (abstract) identity of Arya Stark", not "the body of the girl who grew up as Arya Stark".
Think of it like you're in the army. For most military matters, you will be addressed by your rank (e.g. "cadet"), not your identity. Your name is only used if it is otherwise ambiguous who the person is speaking to.
In contrast to the military, the Faceless Men never encounter a situation where they need to refer to a specific person, as they see themselves as freely interchangeable bodies, which forgoes the need to distinguish between them.
If you have ten identical forks in a drawer, does it matter which fork I give you? No. You'll ask for "a fork", I will give you "a fork", no one (hah!) cares which fork I gave you.
The Faceless Men enforce a "no one" policy, so that they ensure that their followers have no identity, nor a sense of self preservation. A Faceless Man, much like a good soldier, does not shy away from dying for the good of the collective, instead of ensuring his own survival at all costs.
If you're looking for direct answers to your questions:
No one (pun intended).
Jaqen is the (fake) name that "a man" gave Arya when he was pretending to be a normal human. Us humans (the viewer) have therefore always referred to "a man"'s face as Jaqen.
"A man" could have been represented by a different face every time he appears in a scene. His face is not important at all. However, this would be very confusing for the viewers (and would need constant explanation), and therefore "a man" tends to wear the same face, the one that the viewer knows as Jaqen.
Note: The consistent use of the Jaqen-face could also be intended for Arya (in-universe). Arya is not yet a Faceless Man, she does not yet grasp the concept of being no one. Therefore, the Faceless Men might choose to consistently show Arya the same face, simply so Arya can learn.
Notice how she treated the black man who opened the door completely differently than when that man wore Jaqen's face. Arya is not capable of seeing past the face yet.
what he's trying or learning arya?
The philosophy of the Faceless Men.
What is no-one theory?
The philosophy of the Faceless Men. Faceless Men have no identity, they are nothing more than bodies that serve the Many-Faced God.
What kind of service to serve Many-Faced God!
That hasn't really been clarified yet. It is intentionally kept vague. As far as we know, the Faceless Men mostly function as assassins, but their motives for doing so are unknown.
If anything, Arya has been taught to not question the orders she receives. She is not allowed to decide anything for herself (since "herself" is a moot concept to the Faceless Men in the first place).