The word Subject here generally means the part of experience that really seems to be the center part of the person in question, meaning that part that seems to experience the rest of reality, seems to be looking out at things, that part to which things happen, that part that seems to think thoughts, that part that seems to create actions, that part that seems to remember, to know, to control, to do, and to be.
It is generally contrasted with the objects of awareness, which would be all sensations and possible places attention could be put that are not covered in the sense of Subject.
This contrast leads to the concept of Subject-Object Duality.
The sense of the Subject really being The Subject is progressively seen through via insight practices. The complete elimination of the baseline habit of perceiving reality as if the sensations that make up the Subject really were The Subject in a permanent, separate, special way is one way of saying "the end of the path of insight" and is referred to by various names, such as arahatship.