Learn French: Mon manteau est marron.

Mon means "my" and is a possessive adjective, which means it must be placed right before a noun like mon manteau. Le mien means "mine" and is a pronoun, which replaces the noun entirely. You would use le mien if you wanted to say "Mine is brown" (Le mien est marron).

Why doesn't the color adjective marron change to agree with the noun?

In French, color adjectives that are named after things in nature (like fruits, nuts, or flowers) usually do not change to match masculine, feminine, or plural words. Marron comes from the noun for "chestnut", so it never changes its spelling. Even if you had plural coats, it would still be Mes manteaux sont marron.

How can I easily remember that manteau is a masculine word?

A very reliable pattern in French is that almost all nouns ending in -eau (like manteau, chapeau, bateau) are masculine. Because it is masculine and singular, we use the masculine possessive mon.

Do I pronounce the "n" in mon here?

No, because manteau starts with a consonant. The "n" just tells you to make the "o" a nasal sound. You only pronounce the "n" to make a connecting sound (a liaison) when the following word starts with a vowel or a silent H, like in mon ami.