"Their shirt is dirty."
Translation:Leur chemise est sale.
"Son" is "his/her/its". It's only for singular, 3rd-person owners of a single thing. (More singular possessives: http://french.about.com/od/grammar/a/adjectives_possessive.htm)
"Notre/votre/leur" is for plural owners of one of something.
"Nos/vos/leurs" is for plural owners of multiples of something.
For plural owners, there are no gender differences in the possessive. The only differences are in the personal pronoun (1st, 2nd, and 3rd person) and the number of objects being owned.
Namely, "notre/votre/leur" is for multiple owners of one thing, while "nos/vos/leurs" is for multiple owners of multiple things.