Spirit: the nonphysical part of a person that is the seat of emotions and character; the soul
Essentially, by this definition the soul and the spirit are the same thing. I think that is up for debate, but I think it is reasonable to bundle them together when it comes to defining the scope of spirituality. If the spirit and soul are the parts of us which produce our emotions and character, spirituality therefore is a focus on those activities which develop our character and help us manage our emotions.
Whether or not you believe that there is some supernatural element to existence, the activities included in one’s spiritual life invariably fit the above description. We seek to develop our character, to shape our actions, to guide our decisions according to a purpose which reaches beyond our physical existence. Part of spirituality is trying to figure out the rules and traditions which will shape us into virtuous and stable people.