The subject in a simple English sentence such as John runs, John is a teacher, or John was ran over by a car is the person or thing about whom the statement is made, in this case 'John'. Traditionally the subject is the word or phrase which controls the verb in the clause, that is to say with which the verb agrees (John is but John and Mary are). If there is no verb, as in John - what an idiot!, or if the verb has a different subject, as in John - I can't stand him!, then 'John' is not considered to be the grammatical subject, but can be described as the 'topic' of the sentence.
This is a companion discussion topic for the original entry at http://grammarerror.com/tags/subject