The Attributes are three special traits given to certain Trainers and Pokémon when they are born that marks them as a Legendary's chosen one. Beings with Attributes are, for unknown reasons, attracted to each other like magnets and will, for the most part, wind up meeting and travelling together. Each Attribute also has special powers that are exclusive to the Attribute. There are three Attributes: Hearts of Gold, Souls of Silver, and Minds of Crystal.
Finding Your Attribute
There is only one confirmed way for a Trainer to find out if s/he has an Attribute: An affiliated Legendary tells them. There are other ways in the works, but they are not official yet.
If you want one of your Trainers to have an Attribute, all you've gotta do is ask in the Discussion thread, and we'll probably say yes. They are, after all, attracted to each other, and the main characters are PlotMagnets anyway. HOWEVER: Your Trainer is completely unaware of this fact and can only find out using any of the described methods. This doesn't mean that you can have a Legendary pop out of nowhere and tell your character that s/he has X Attribute, and your character instantly and completely understands exactly what the Legendary means and exactly how to use it in an overpowered way. Treat it like any real person would react to such shocking news, an have them gradually get used to the concept and start learning what new stuff they can do.
It is also completely okay for your Trainer to not have an Attribute. They don't have to have special powers to be awesome.