In ED the universe is persistent and dynamic, hence the need to be online regardless of the mode you're playing in. Your actions, such as trading, exploring, laserising other ships, banging into things and all that jazz, will be constantly used to update the universe, along with similar actions by both NPC's and players.
The three modes of play - Open, Group, And Solo - dictate who else will populate your experience of the universe, and you can switch between the three freely and at almost any time.
Open play will give a universe populated with both NPC's and players, so in a sense this is the multiplayer experience. However, ED works on an instancing basis, which means that your experience of a system may not include all the players who are currently in that system. If there are, say, 50 players in your particlular system, your instance may only include 10 of those players, plus NPC's. The other 40 players will be in other instances within the system. This can make it tricky to match up with a friend as you both have to be in the same system and the same instance to see each other. There are developments on the way to address this but for now, that's how it is.
Group play will populate the universe with NPC's and any other players who belong to the same group that you're playing in. This is also multiplayer but the players pool will be much smaller and unless you arrange beforehand to meet up, the likelihood of seeing a player character is considerably less. There will still be many NPC's around though.
Solo play limits other ships to NPC's only, with no player characters.
As mentioned earlier, you can switch between different play modes freely, and so you can use this mechanic to help you out if need be. Sometimes in Open play you will find long queues to dock at Outposts, switching to Solo play tends to remove these queues to allow docking - useful if you're up against a mission timer that is running out! Another use for this ability to switch is if you find yourself trapped in a station with a player character outside who generally has little else to do but shoot you up every time you try to leave (not happened to me but I have heard of such stories) - switch to Solo mode and this problem is removed while you make good your escape.
Hope this explains things for you!
PS - buy the game, it's truly brilliant.