The essential elements of thought, the subject and its predicate are all that is needed to compose a sentence; they initiator of the action and the action itself. Because of this very essential relationship, other grammatical rules follow with necessity such as Subject/Verb Agreement. That is, the subject must agree with its predicate in person and number. Now, since Modern English has all but dispensed with conjugated endings, there are few changes to be made. However, some remain such as in third-person singular when the regular verbs have an -s added to them. e.g. He/she talks.
From my learning Latin, I find it helpful to think of the subject as being the person or thing doing the action. In the sentence, "Karen is sitting on the chair," Karen is clearly the person doing the action and so she is the subject. This is a useful way for me to think through and identify what the subject is in sentences.