Definition of commutative and non-commutative algebra and algebra isomorphism

$mathbb Clanglelangleldotsranglerangle$ denotes the ring of power series in noncommuting variables.

$mathbb C[[ldots]]$ denotes the ring of power series in commuting variables.

Frankly I don't know what else could be meant by "$W$ is an isomorphism between $A$ and $widehat{A}$" other than "$W$ is a bijective ring homomorphism between $A$ and $widehat{A}$".

That's all I can say given the limited context.

I do understand the meaning of the ideal.

"$mathcal I$ is the ideal generated by the commutation relations of the coordinate functions"

I would take that to mean the ideal generated by all the relations necessary to describe the difference between $hat{x}_ihat{x}_j$ and $hat{x}_jhat{x}_i$. I don't know what that is in your particular case (lack of context again). If you include $hat{x}_ihat{x}_j-hat{x}_jhat{x}_i$, that says the two coordinate functions commute. But it could also be something like $hat{x}_ihat{x}_j-hat{x}_jhat{x}_i-ihbar$ depending on their relationship.