Isn't the reason that Sunnis and Shiites fight over the descendency of the prophet Mohammed?

Can anyone explain this with a bit more detail. If they are all Muslim (or Moslems, whichever is politically correct) why does it matter so much? What does the lineage signify? What it not be logical to conclude perhaps some type of agreement instead of literally slaughtering each other? Does the teachings of Islam justify all of this bloodshed, or is it more or less the act of religious 'extremist'? To which 'side' do 'other' Muslims side? Sincere curiosity speaking, any thoughts or information would be appreciated.