Solo is just you and a universe of NPCs, but what you do in the game does have an effect on the economy etc. You are just invisible to all the other human pilots, and they to you.
Open is you and the NPCs, plus everyone else who is also in Open. Combat vs. everyone is allowed and friendly fire can be taken, so in the final analysis it is essentially PvP, but not like most PvP "arenas".
Private Group is half way between the two - it's like solo, but with a bunch of buddies - all you ever see is chosen players.