"She puts her coat on."
Translation:Elle met son manteau.
More specifically, which you use depends on the gender of what you are talking about, UNLIKE in English where it depends on the gender of the person (his/her). For example, "son manteau" here, since manteau is masculine, while "sa pomme" would be used because pomme is feminine.
The exception is in front of a vowel or mute-h. Then the masculine is used, to avoid having two vowels next to each other. For example "son orange" is used despite orange being feminine, in order to keep the flow and to sound nicer.
It's not reflexive because the verb "mettre" isn't a reflexive verb! It is an idiomatic pronominal verb however, which means that while "mettre" means "to place, to put", "se mettre à" means "to begin to". For a more thorough understanding of pronominal verbs (including reflexive verbs), I recommend taking a look at this website: http://french.about.com/od/grammar/a/pronominalverbs.htm