mcc you are quite correct, there are several aspects to playing the game.
1. play the storyline through solo (PvE)
2. there are missions(strikes) that you can play with a team of 3 (matchmaking by sony)
3. there are missions that can be played with teams of 3 or 6 that are not formed by matchmaking so you need a team of friends to play them (very difficult)
4. there is a part of the game called Crucible which is PvP which is formed again by matchmaking but is terribly difficult
And yes, you need to be hooked up to PSN and all data is kept at the Bungie server online.
Not for everyone but my sons and I love playing it. Can be a bit addictive