The word 'subject' needs care as it has a particular - and very important - meaning that is quite distinct to grammar and which is different from the meaning you probably usually use. Grammatically speaking, the subject (S) is an element of a clause. It is the word or phrase that the clause tells about - and is usually a noun or noun phrase. In the sentence, 'I gave him a present', 'I' is the grammatical subject and 'gave' is its associated verb in the sentence (in the past tense). In the simple sentence, 'The quick brown fox jumped over the lazy dog', the subject is 'The quick brown fox'. This is a noun phrase that has as its associated finite verb , 'jumped'.
Most English sentences need a subject but sometimes this can be one of the small words (called pronouns) 'it' or 'there'. This type of subject can be tricky to recognise as proper subjects.
Some typical word orders of simple declarative sentences are: SV (subject-verb), SVO (subject-verb-object), SVC (subject-verb-complement) or SVA (subject-verb-adverbial).
Some types of verb transfer their action from their subject onto something else (the thing receiving the action of the verb is called its object). These are called transitive verbs. In the above sentence, the verb 'gave' is transitive as action transfers to the object, the noun 'a present'.
Verbs are called intransitive if they do not transfer action, but, instead, act to tell what their subject is doing, e.g. 'He is working.', 'It died.' Some verbs can be either transitive or intransitive according to their usage in the sentence, e.g. 'He is singing.' (intransitive) and 'He is singing a song.' (transitive).
A few special verbs (stative verbs) have no sense of direct action but, instead, act to make a statement about their subject's state of being. These verbs are called copular or linking verbs, e.g. He seems ill, She is clever, he was a criminal, it appears dark, etc.. The word that follows a stative verb has no action passing on to it so it cannot be called an object; instead, it is termed a complement.
Confusingly, Some verbs can take two objects:
'I gave Sally a present.' (i.e. 'I gave a present to Sally')
In this type of sentence, the object is 'a present' (= the thing given; this is called the DIRECT OBJECT); but there is a second 'object' - the 'receiver' of the direct object. This is termed the INDIRECT OBJECT. Notice that all sentences of this type can be re-written as shown using the word 'to'.