Why does nuestro agree in gender but mi does not?

Historical accident. The first and second person singular forms (mi, tu) and the third person form (su) only mark number, not gender - mi casa and mi libro both use mi. But nuestro and vuestro inherited the full Latin agreement system and still mark gender as well as number, so you get four forms: nuestro / nuestra / nuestros / nuestras. Treat them as the exception, not the rule.

Does su mean his, her or your?

All three, plus their and your-formal. Su is the most ambiguous word in beginner Spanish. Su casa could mean his house, her house, your house (usted), their house, or your house (ustedes). Context usually disambiguates, and when it doesn't, Spanish speakers fall back to the long form: la casa de él, la casa de ella, la casa de ustedes. If you find yourself writing su three times in a paragraph, it's often clearer to rewrite with de + pronoun for the ambiguous ones.

Spanish Possessives

Spanish short-form possessives sit before the noun (mi coche, tu casa, nuestros amigos) and agree with the thing possessed, not the possessor. That last bit is the rule English speakers forget: nuestras hijas (our daughters) uses the feminine plural because hijas is feminine plural, regardless of who the parents are.

The full table

Possessor	Singular	Plural
yo (my)	mi	mis
tú (your, informal)	tu	tus
él / ella / usted	su	sus
nosotros (our)	nuestro/nuestra	nuestros/nuestras
vosotros (your, pl)	vuestro/vuestra	vuestros/vuestras
ellos / ellas / Uds.	su	sus

Note the split: mi, tu, su only mark number (mi/mis, tu/tus, su/sus). Nuestro and vuestro mark both number and gender (four forms each). There's no logic to this beyond Latin inheritance - memorise it as the pattern.

Agreement with the noun

The possessive matches the thing possessed.

mi libro (my book - masculine singular)
mis libros (my books - plural)
nuestra casa (our house - feminine singular)
nuestros amigos (our friends - masculine plural)
vuestras hermanas (your sisters - feminine plural, vosotros possessor)

The gender of the possessor doesn't matter. A father and a mother both say mi hija (my daughter) and mi hijo (my son), because the agreement is with the child's grammatical gender, not the parent's.

The su problem

Su covers his, her, your-formal (usted), their, and your-plural-formal (ustedes). It's the most overloaded pronoun in the language. Su coche could mean any of:

his car
her car
your car (formal singular)
their car
your car (formal plural)

Context usually fixes it. When it doesn't, Spanish reaches for the de + pronoun workaround: el coche de él, el coche de ella, el coche de ustedes. This is grammatical, common in speech, and the right move whenever su would be ambiguous.

Worked examples

Mi madre vive en Madrid. (My mother lives in Madrid.)
¿Dónde están tus llaves? (Where are your keys?)
Nuestra casa es pequeña. (Our house is small.)
Sus hijos estudian en Londres. (Their / his / her / your children study in London.)
Vuestros amigos son simpáticos. (Your friends are nice. Spain only - vosotros possessor.)

Body parts and clothing

A quirk worth flagging: Spanish often drops the possessive for body parts and clothing, replacing it with the definite article plus a reflexive or indirect object pronoun.

Me duele la cabeza. (My head hurts. Literally: the head hurts to me.)
Se lavó las manos. (He washed his hands.)
Ponte la chaqueta. (Put on your jacket.)

Saying me duele mi cabeza is grammatical but sounds foreign. The article-plus-pronoun construction is the native default.

Long-form possessives (briefly)

Spanish also has long-form possessives - mío, tuyo, suyo, nuestro, vuestro, suyo - which go after the noun or stand alone as pronouns.

un amigo mío (a friend of mine)
ese coche es tuyo (that car is yours)
el libro es suyo (the book is his / hers / theirs)

These are Higher-tier and lived through later. For the Foundation curriculum, the short forms above carry 95% of the load. The intermediate Spanish grammar page covers the long forms in full.