You always use the article (il, la, lo etc.) before possessive adjectives (mia, mio, tuo etc.) unless you're referring to a member of the family or it's a set expression. 'La mia macchina', not 'mia macchina'. 'Mio fratello', not 'il mio fratello'. It also applies to members of somebody else's family; 'Alessandro è suo fratello', not 'Alessandro è il suo fratello'. However, you say 'mio dio' or 'dio mio', not 'il mio dio' as this is a set expression.
"Il mio" would not be wrong in this case, but it has a slightly different meaning. "Il cappello marrone è il mio" implies you're identifying a specific hat as yours in a group ("The brown hat is the one that's mine"). "Il cappello marrone è mio" just shows your possession of the hat in general. English doesn't distinguish between these two and uses "is mine" in both cases.
I am still rather confused about when to use the pronoun in front of the prepostion. I get the explanation of 'il mio fratello' being wrong because you don't use it for family members etc ( as explained below by Yuujen) but I am still wanting to find out why il cappelo marrone e il mio is wrong.
I'm not surprised. I've never heard anyone describe a hat's color (or anything else for that matter) as "sweet chestnut" (and get away with it), (unless it's of course a sweet chestnut), regardless of what DL says.. Oh, you'll find it in one of those upscale mail-order catalogs, like Territory Ahead, that have come up with the most ridiculous saccharin sounding names for colors to sound hip, but you won't hear it in common street talk, so why use it? Ok, so maybe you're a rancher with a chestnut mare that's just a sweetie, well then that's fine, but forget about it for a hat, coat, gloves, socks, and especially underwear, unless you've got street cred in MMA. (Just a suggestion).