I heard Destiny is a "persistent world" what does that mean?

Kotaku notes that in Destiny your Guardian  live in a persistent world. What is a persistent world?

It's a world that is "constantly being affected by not just the passage of time, but by other players. As you enter certain areas of the game, the server will register which of these are public domains/combat areas. If you stumble into one, it'll automatically, and invisibly, search for another gamer to matchmake you with."

It means other Guardians may unexpectedly drop in to your own campaign to help you out. 

"So as you play, the server should seamlessly integrate another player into your game. As you're scouting a safe path around the dangerous Cabal, for instance, you might bump into another player. So even though you're playing out your personal story, a lot of your time is spent in shared adventures."

This is a new twist for the console platform - it's taking the concept of co-operative play and adding a twist. Destiny, will still be your story but other Guardians' energies and efforts will have an effect on yours. 

To that end, Destiny needs to be played with an always on internet connection as it was obviously built for social and cooperative play. Clearly, Bungie expects social gaming and interaction at this level to be the way of the future.