You could say that there is only one game mode in the Elite: Dangerous - it is a sandboxish free roaming. It can be somewhat compared to the Minecraft or DayZ: You start the game and right away you are on your own. Everything is dependent from you and the enemies or friends you will encounter on your way.
Elite: Dangerous went with the way of the Evochron MerPricery, where the progress is being transmitted between the single and online multiplayer. There are three modes to play: Solo Play, Private Group and Open Play.
In the Solo Play you are playing by yourself, but the economic and political system are common and shared between other players, thus making the game similar to other MMOs. This is why you cannot play the game without an internet connection. In practice it would look like this:<br>Imagine that you are interested in a particular resource that you've just seen on the other station. You're coming back there and it turns out there is no more of this resource as probably the other player earlier came up with the same idea of buying it.
Private Group is the mode is for everyone who wants to play online but in the hermetic group of their own friends in their private group. These friends can leave or join the group at any time (so be careful who you invite). There are no more differences between this mode and the solo mode.
Open Play is simply a MMO where everyone minds their own business. It may result with meeting new friends and acquaintances in the common operations. It is also a bit risky as there are no limitations concerning attacking other players. The only limitation is the law which applies in the particular areas and its representatives. Each misdemeanor is noted down and the head is being exposed for the hunting of bounty hunters. Such wrong doers are marked and the player and the AI law enforcers can attack them without any consequences. The Open Play mode is definitely for those who are not afraid of the risk.