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Possessives agree with the thing owned

In English, we say "his car" if it belongs to a man and "her car" if it belongs to a woman. The possessive word changes based on the person who owns the object. French does the exact opposite. French possessives change based on the gender and number of the object being owned, completely ignoring who owns it.

Because of this, words like son and sa can translate to "his", "her", or "its" depending entirely on the context. If you want to say "his sister", you must use a feminine possessive because the word for sister is feminine. If you want to say "her brother", you use a masculine possessive because the word for brother is masculine.

Word list
le leurmA1 theirs agrees with object: le leur, la leur, les leurs le mienmA1 mine agrees with object: le mien, la mienne le nôtremA1 ours agrees with object: le nôtre, la nôtre le sienmA1 his agrees with object: le sien, la sienne le sienmA1 hers agrees with object: le sien, la sienne le tienmA1 yours agrees with object: le tien, la tienne le vôtremA1 yours agrees with object: le vôtre, la vôtre leurA1 their agrees with object: leur before singular noun, leurs before plural noun monmA1 my agrees with object: mon before masculine or vowel, ma before feminine consonant notreA1 our agrees with object: notre before singular noun, nos before plural noun safA1 his agrees with object: sa before feminine noun (unless starting with a vowel) sonmA1 his agrees with object: son before masculine noun or feminine noun starting with a vowel tafA1 your agrees with object: ta before feminine noun (unless starting with a vowel) tonmA1 your agrees with object: ton before masculine noun or feminine noun starting with a vowel votreA1 your formal; agrees with object: votre before singular noun, vos before plural votreA1 your agrees with object: votre before singular noun, vos before plural le vôtremA2 yours formal; agrees with object: le vôtre, la vôtre
Possessive adjectives

Possessive adjectives are the little words placed before a noun, like mon (my), ton (your), or sa (his/her/its). When using these words, you must look at the noun that comes right after them to decide whether to use the masculine or feminine form.

This is especially tricky with the third person. Both "his" and "her" are translated as son before a masculine noun and sa before a feminine noun.

Marie parle avec son frère.
(Marie is talking with her brother.)
Paul a perdu sa clé.
(Paul lost his key.)
Paul cherche sa veste, et Marie cherche son manteau.
(Paul is looking for his jacket, and Marie is looking for her coat.)

Possessive adjectives for plural owners, like notre (our), votre (your), and leur (their), do not have separate masculine and feminine forms for singular objects. They only change if the objects being owned are plural.

Ils aiment leur voiture.
(They love their car.)
Possessive pronouns

Possessive pronouns replace the noun entirely. They include words like le mien (mine), le tien (yours), and le sien (his/hers). Just like possessive adjectives, these pronouns must match the gender and number of the thing they are replacing.

For example, if you are talking about a house (feminine), you will use feminine pronouns like la mienne or la sienne, regardless of whether the owner is a man or a woman.

Cette maison est la sienne.
(This house is his. OR This house is hers.)
Mon vélo est rouge, le tien est bleu.
(My bike is red, yours is blue.)
Ces livres sont les nôtres.
(These books are ours.)
Watch out: The vowel exception

There is one major exception to the gender agreement rule. To make pronunciation smoother, you cannot use the feminine words ma, ta, or sa before a noun that starts with a vowel or a silent H.

Instead, you must use the masculine forms: mon, ton, and son. The noun itself remains feminine, but the masculine word is used purely for the sound.

C'est mon amie.
(She is my friend.)
Elle adore son école.
(She loves her school.)
Summary

When choosing a French possessive word, always look at the thing being owned. The possessive must agree in gender and number with the object, while the gender of the owner does not change the word you use.