Here are some definitions of 'subject'.
: a noun, pronoun, or group of words substituting for a noun, that engages in the main action of the sentence or is described by the sentence. (from the Allyn and Bacon Handbook).
: the part of a clause about which something is predicated. ... A subject is the person, thing, or idea about which something is said or asked. Locating a subject therefore involves locating its accompanying verb. (from the Random House Handbook).
: a noun or noun equivalent about which the sentence is predicated and (esp.) with which the verb agrees. (from the Shorter Oxford English Dictionary).
To identify the subject, you must identify the verb and vice versa. This may seem unsatisfactory but there it is. Also consider that the subject and verb are the essential components of any independent or dependent clause: you can delete all other parts and still retain some core meaning but you cannot delete the subject and verb. Don't ask me to define 'core meaning'. So, the subject and verb provide the essence of the story.
The subject can be categorized as simple, meaning the smallest unit that acts as a subject, or whole, meaning the smallest unit plus related modifiers that acts as a subject.
Example: Separation of means in this analysis requires homogeneity of variances.
simple subject: Separation
whole subject: Separation of means in this analysis
Also see clauses.