Objects are noun phrases that are included in the predicate. They are the things being acted upon by the verb. For example:
Susan bought the gift.
The dog caught the ball.
The boy spilled the milk.
In each of these sentences, there is a direct object in the predicate. It is the thing being acted upon; for example, the gift is the thing given by Susan.
It's also possible to have an indirect object that includes more information about the person or thing towards which the action is directed. For example:
Susan bought him the gift.
He wrote them a long letter.
The man gave her an ultimatum.
In the sentence "Susan bought him the gift," "him" is an indirect object because he is not the item bought at the store, but rather the person for whom the action was completed.
Complement
In predicates that use linking verbs rather than action verbs, items following the verb are known as complements. Complements modify the subject by describing it further. For example:
I am a teacher.
The cat was the laziest creature.
The woman seems smart.
In these cases, the words following the linking verb describe the subject, whether they are nouns, noun phrases or adjectives.
To write a complete sentence, you must include at least a subject and a predicate. If you only include one of these, you will have only a sentence fragment, which is grammatically incorrect.
Modifier
Modifiers are words or phrases that describe parts of the sentence by adding additional information. Adjectives modify nouns and pronouns, while adverbs modify verbs, adjectives, and other adverbs. It is possible for parts of speech to do this work alone in the sentence in either the subject or the predicate. For example:
The blue boat sank.
The boat slowly sank.
In the first example, the adjective modifies the subject, but in the second example, the adverb modifies the simple predicate.
Adverbial
Not all modifiers are single words. Sometimes they are groups of words that work together. When these words are in the predicate and explain how, when, where or why the action was performed, they are known as an adverbial. For example:
She exercised in the morning.
The boy reached out to the teacher for help.
I skipped school because I was sick.
Each adverbial above modifies the verb, therefore performing the function of an adverb in the sentence. The first two examples are adverbial phrases - groups of words that function as an adverb but don't contain a subject and a verb. The final example is an adverbial clause, which performs the same function but does contain a subject and a verb. Adverbial clauses are dependent clauses and are not complete sentences on their own.
Modifiers are optional when it comes to writing sentences since they merely give additional information and are not required the way a subject and verb are. Note that complements are considered to be a type of modifier as well.