Firstly let me state that we are talking about a complement not a compliment. a compliment is when someone says something nice to you, and a complement is when something is added to something else. In grammar a complement is added to a verb to aid or complete its meaning.
A complement is a word or group of words that completes the meaning of the verb. There are four kinds of complements: direct objects, indirect objects, object complements, and subject complements.
Direct Objects. These answer the question what? or whom? after an action verb. The subject of the sentence usually performs the action indicated by the verb. That action may be directed toward or received by someone or something-the direct object. Direct objects are nouns, pronouns, or words acting as nouns; and they may be compound. Only transitive verbs have direct objects. Examples of direct objects are: “Estella sold her ‘computer’.” [Estella sold what?] “Tamara watched the professor.” [Tamara watched whom?] “Estella sold her ‘computer’ and ‘radio'” [Estella sold what?]
Indirect Objects. These answer the question ‘to whom?’ ‘for whom?’ ‘to what’ or ‘for what?’. A sentence can have an indirect object only if it has a direct object. Two clues can help you identify indirect objects. first, an indirect object always comes between the verb and the direct object. Eg. The owner gave ‘us’ a discount. [the owner gave a discount to whom?]
Object Complements. These answer the question ‘what?’ after a direct object. That is, it completes the meaning of the direct object by identifying or describing it. Object complements occur only in sentences with direct objects and only in those sentences with the following action verbs or with similar verbs that have the general meaning of ‘make’ or ‘consider’. An object complement usually follows a direct object. It may be an adjective, a noun, or a pronoun. Eg. “The bonus made Susan ‘happy’.” [adjective]“I named my dog ‘Sadie’.” [proper noun]“Our cat considers that pillow ‘hers’.” [pronoun]
Subject Compliment. This follows a subject and a linking verb and identifies or describes the subject. There are two kinds of subject complements: ‘Predicate nominatives’ and ‘predicate adjectives’.
A predicate nominative is a noun or a pronoun that follows a linking verb and points back to the subject to rename it or to identify it further. Eg. “Cellists are ‘musicians’.” “The soloist for this concert is ‘someone’ from Dallas.” “My favorite singer is ‘he’.” Predicate nominatives are usually found in sentences that contain forms of the linking verb ‘be’. A few other linking verbs as well (become and remain) can be followed by a predicate nominative. E.g. “Alexis remains an ‘admirer’ and a ‘friend’.” “That class became a ‘challenge’ for me.
A predicate adjective is an adjective that follows a linking verb and points back to the subject and further describes it. E.g. “My sister is ‘generous’.” [the adjective, generous, points back to and describes more about the subject ‘sister’]“Some doctors are ‘compassionate’.” [The adjective compassionate, points back to the subject ‘doctors’]