The term “possessive” in grammar often refers to actual possession as we normally think of it. When we talk about “Sam's phone,” we are using a possessive indicator ('s) to say that we are talking about a phone that is possessed by Sam. An “s” preceded by an apostrophe (') is the device we use to make ordinary nouns possessive in English. However, once we make a noun possessive and place it in front of another noun, it stops being a noun and becomes an adjective. In this case, the possessive form “Sam's” describes, or “modifies,” the noun “phone.” It is a possessive adjective.
Grammar somewhat stretches the idea of possession, however. First of all, it allows things to possess other things—for example, the city's shopping district, the atom's nucleus, the film's director. It also allows relationships that are not exactly possessive in nature: the woman's request, the room's temperature, the boss's retirement, the country's largest city. So sometimes the meaning of the possessive form is not strictly possessive but describes a relationship of one thing being in close association with another thing--one thing may be a part of another thing, for example, or may be controlled by, created by, or located in another thing.
Some very important words in English are the possessive adjectives that are related to personal pronouns. These are the words “my,” “his,” “her,” “their,” “your,” “our,” “its,” and “whose.” These words, of course, are used to indicate possession in both the usual and “stretched” senses of the word, as in “my jacket,” “my job,” “my idea,” “my misunderstanding.”
Another function that possessive adjectives have is to show the agent (the doer) of the action described by a gerund—a verb in its “-ing” form functioning as a noun. So, for example, in the sentence “I don't like his coming home so late,” the doer of “coming home so late” is a certain male person, and his action of coming home so late is something that the speaker does not like.
In sentences similar to the example above, it is quite common for speakers to use object pronouns (“me,” “him,” “her,” etc.) or ordinary nouns instead of possessives: Instead of saying “I don't like his coming home so late,” many people would say “I don't like him coming home so late.” Instead of saying “They anticipated the doctor's saying that,” many people would say “They anticipated the doctor saying that.” Their intended meaning is most likely the same as if they used the possessive form in these instances.
Strict prescriptive grammarians might insist that because the forms “coming” and “saying” in the above sentences appear to be nouns (verbal nouns), they must be preceded by an adjective, and therefore it is an error to say “I don't like him coming home so late.” From this point of view, “him” is simply wrong and only “his” would be correct. Although there is good logic in this, use of the object pronoun may be surpassing the use of the possessive form in such sentences.
One reason that people have a strong tendency to use object pronouns and regular nouns rather than possessive forms in these types of constructions is that there is a similar construction in which the object form (e.g., “him”) is the correct and ordinary form preceding a verb with “-ing.” This is when it is truly the sole object of the verb and this object just happens to be doing something that the speaker wants to mention. For example, the sentence “I saw him coming into the house” is basically the equivalent of saying “I saw him as he was coming into the house” or “I saw him, and he was coming into the house.” The sentence might answer the question “What was he doing when you saw him?” It is also possible and correct to say “I saw his coming into the house” using a possessive form, but it would be less common and not of exactly the same meaning. This sentence answers the question “What did you see?” and the answer (the complete direct object) is “his coming into the house.” It is the equivalent of “I saw the coming into the house by him” or “I saw his act of coming into the house.” Technically, the “-ing” form following the object pronoun is a present participle, and the “-ing” form following a possessive adjective is a gerund (a verbal noun).