What do possessive case of nouns do to sentences?

A possessive noun indicate ownership or possession.

the car of my mother = my mother's car

the teacher of our class = our class's teacher

the coats of the children = the children's coats

the covers of the books = the books' covers

A possessive noun indicates origin or purpose.

children's shoes; not shoes belonging to children, shoes intended for children

ladies' room; the room isn't owned by a group of women, it's a room intended for their use

Shakespeare's plays are not possessed by Shakespeare, they're plays by Shakespeare.

today's newspaper, today can't own or possess, the newspaper originated today