This one is pretty obvious: complete all story missions on Normal (or Heroic/Legendary). There are 40 story missions total, and ODST missions are not required.
This achievement is tied directly to the individual achievements that unlock after completing each mission. If you have completed each mission and this one does not unlock, ignore what the in-game menu says and check your other achievements to see which story mission achievement did not unlock and replay that mission.
This might be obvious to some, but mine did not unlock despite having played all missions, and the in-game menu stating that every mission had been completed on normal or higher. I re-played several missions before I thought to check the achievements, and found which mission did not unlock.