Destiny is an always online shared-world shooter. Players can play alone but will get the best experience with others. Even though Destiny is a multiplayer focused game, you can play a very large part, almost the entire story mode, alone if you so desire. However, there are some parts where you need five people to start a mission for example.
Take a look at Bungie where they discuss feedback from the Beta and changes to expect in the final game. Check out the Sneak Peek Into Destiny's Competitive Multiplayer.
