Main functions
1.- Cuyo is used to give a definite value to a noun to which it always preceeds and with which matches in gender and number:Ese es el chico cuyo padre es un reconocido arquitecto.
That's the guy whose father is a well-known architect.
Esa es la falda cuya cremallera se rompió.
That's the skirt whose zipper broke.
2.- The main function of cuyo and any other forms is its possessive meaning. Cuyo/a/os/as express possession by the noun that preceeds it with which it does not match in gender and number (the possessive cuyo matches in gender and number with the noun that goes after it):
Aquella casa, cuyas ventanas están rotas, está puesta a la venta desde diciembre.
That house whose windows are broken is on sale since December.
3.- There is some situations where cuyo has a relative value instead of a possessive value. En cuyo caso, por cuya razón, con cuyo objeto, and a cuyo fin are idioms that have that relative value:
Deja papel y bolígrafo cerca del teléfono en cuyo caso no encontrarás ninguno cuando lo necesites.
Leave a pen and paper near the telephone in which case you will not find any when you need it.
Important Aspects
A.- The use of cuyo is not compatible with demonstrative adjectives nor the indefinite todo, but it is compatible with numerals:Estos cuyos hijos (INCORRECT)
María, cuyo segundo hijo está estudiando en una universidad de Estados Unidos, está haciendo un gran esfuerzo para pagar la matrícula.
Maria, whose second chldren is studying in a U.S. university, is making a great effort to be able to pay the tuition.
B.- The use of cuyo is not a substitute for que + possessive adjectives su, sus:
Devolví ese libro que sus páginas estaban manchadas. (INCORRECT)
Devolví ese libro cuyas páginas estaban manchadas. (CORRECT)
I returned the book whose pages were stained.
C.- The use of cuyo in front of feminine nouns that begin with a stressed a or ha does not change the feminine for cuya to cuyo as it happens with the definite article:
El agua está fría (Not La agua está fría)
El mar Mediterráneo cuya agua está siempre caliente en verano tiene unas de las mejores playas del mundo.
The Mediterranean sea whose waters are always warm during the summer has the best beaches in the world.