The subject of a sentence is the center of attention of a sentence. A subject is almost always a noun phrase. It can be a concrete thing or it can be an abstract idea or topic that is being talked about. Subjects are typically found at the beginning of the sentence and often start the sentence. You can always determine the subject of a sentence by finding the verb and asking "who" or "what" before the verb. The answer to the question will be the subject of the sentence.
A sentence predicate is the other half that completes a sentence and generally describes what is being said about the subject. A predicate starts with a verb that expresses action towards the subject or a state of being about the subject. You will always find the predicate after the subject in a declarative sentence. If you know the subject, you can ask "what about [insert subject]" and the answer will be the predicate.
Subjects can be built around topic or noun phrases; however, they can always be boiled down to one noun or pronoun. That one noun or pronoun is the simple subject. For example, in the sentence, "Four warm cups of coffee spilled on the table," the subject is "four warm cups of coffee." The subject can be stripped of the prepositional phrase "of coffee" and the adjectives "four warm" to show the simple subject, "cups."
The predicate can be one or more verbs and verb phrases; however, like the subject, it can always be stripped down to one verb. The predicate in most simple sentences generally includes the rest of the sentence after the subject. For example in the sentence, "Aishah brought four warm cups of coffee into the room," the predicate would be everything after the subject "Aishah." The simple predicate is the verb "brought," without the adjectives or prepositional phrases.