This is particularly hard for English speakers since in English, you choose the possessive according to the gender and number of the subject: for a man, his dog(s), for a woman, her dog(s). The gender or number of the dog does not matter. In French, it's the other way around... So, when talking about a man, you write "sa maman", it looks weird, but it is correct in French. You have to change the way you think.
Our possessive adjectives have a double entry logic :
1 - first you choose the correct possessive form according to the subject
je - mon, ma, mes (my)
tu - ton, ta, tes (your)
il / elle - son, sa, ses (his / her)
nous - notre, nos (our)
vous - votre, vos (your)
ils / elles - leur, leurs (their)
Note: the possessive for "on" depend on the meaning of on. It can be "son, sa, ses" or "notre, nos".
2 - Then you choose the correct gender and number according to the object (the word that comes AFTER the possessive adjective)
My dog can be "mon chien" for a masculine dog, or "ma chienne" for a feminine dog, or "mes chiens" if I have several dogs. The speaker gender and the number of people speaking do not affect the gender/number of the possessive adjective.
In other words, let's say that we are talking about one male dog. "Anne" would say "mon chien" (not ma chienne), because she has one male dog (the fact that Anne is a woman doesn't count at all).
"Sophie and Anne" would say "notre chien" (not nos chiens), because the both of them only have one dog, and again, what counts is the quantity of dogs, not that Sophie and Anne are plural.
If you can build reflexes on this double entry logic, you'll master the possessive adjectives.
Watch out for ses and leurs
This is a mistake I hear too often: "ses chiens" to say "their dogs".
Ses chiens = his dogs - refers to a 3rd person singular ; he or she. Your choice is among sa, son, ses. This person has several dogs, hence "ses".
Leurs chiens = their dogs - refers to 3rd person plural - they. Your choice is among leur and leurs.
Watch out for the "mutant" form!!!
When a feminine word starts with a VOWEL or a mute H, you won't use ma, ta, sa BUT mon, ton, son (the masculine form) for a reason of pronunciation - to avoid a clash of vowels
Dont' say "ma amie" but say "mon amie" with a strong liaison [mon Namie]. "Ma amie" is a jaw breaker. So it's a lazy mouth rule :-) Same goes for "ton auto" (your car), "son école" (his school).
French possessive adjectives never take elision
M', t', s' stand for me, te, se, never possessive adjectives.