Hi Katie here you have the explanation about these words:
Marrón is used as the colour when you want to describe any object as a car, a plant so let´s say to describe things. El coche es marrón.
Moreno, is used when you want to describe the colour of the skin of one person (Ella es morena o ella está morena).
Castaño: is used when you want to describe the colour of the hair of one person (Ella tiene el pelo castaño). This word is also used for the colour of the eyes (Tiene los ojos castaños) but here for the eyes you can also use marrón (tiene los ojos marrones)
If you have any other doubt don´t hesitate in keep in touch.
regards,