How presidential powers clearly tied to his/her responsibilities

Presidential Roles include:
Chief of State
Chief Executive
Chief Diplomat
Commander-in-Chief
Legislative Leader 
Chief of Party
Guardian of the Economy

Presidential powers all stem from being able to carry out certain tasks related to these roles and responsibilities. While the President may not have the direct power to control or do things his powers allow for influence that can often be felt decades later. 

Chief of State - often related to being that "inspiring" example of America, patriotic speeches, welcoming foreign guests; everything to help pump up and show why America is so great; show pride
Chief Executive - run the "business" of government and the many employees that work under him, conduct meetings to help problem solve any possible issues. What and how to enforce laws
Chief Diplomat - Determines the tone and what is said to foreign governments through the many diplomats around the world. Helps create and direct foreign policy 
Commander-in-Chief - decides where troops go and what weapons will be used. Generals take orders from President
Legislative Leader - influence laws (he can't make them) veto and sign laws
Chief of Party - help members of the party get elected/re-elected; campaign for reelection if possible; support those that have supported him 
Guardian of the Economy - address and try to help stop any problems, if possible. Help the economy to run smoothly; monitor things like unnemployment and taxes