I imagine missions will be similar to older elites plus whatever else they dreamed up.
Transporting items, documents and passengers to certain planets. Possibly being attacked on the way if that thing you are transporting has been desired by somebody else.
A mission to intercept somebody transporting items, documents and passengers.
Imagine being offered a contract to transport some unpopular senator to Freeport before a certain time, while another few players had been offered a contract to kill the guy. You have to get from A to B without them getting you first. But luckily someone, maybe yourself, hired somebody to escort you.
Missions to gather intel on certain planets or people or space anomaly something.
Maybe the people paying the contract dont want any attention drawing to themselves or anybody finding out you are doing this for them so doing it undeteced might be part of the deal.
Killing a certain pirate/Bounty hunter/Trader/High profile target/whatever.
Hunting down a ship with limited knowledge of its location before it escapes into deep space.
Distress signals. A hired gun to fight in wars. Finding stuff.
Escort missions, defending a spacestation from attack. Destroying a spacestation, structure.
Missions from one faction against another. This would build your reputation with one faction but could damage your rep with another.
Some missions may require you to break the law and have entire police forces on your back. Upsetting certain factions could make all them hostile and attack on sight. Some missions may cause you to get a huge bounty on your head and have the entire Elite dangerous forums on your back.
Missions will range from basic, Get to A to B. To Damn right suicidal assassinating a heavily guarded police chief.
Missions will be offered based on skill, reputation, and your popularly or trustworthiness of the persons offering the mission.
Nobody is going to offer you any sort of mission that requires a bit of skill or responsibility if nobody even knows who you are and you barely made it out of the dock door on your first outing into space.
Missions could be offered at spacestations, or possibly on the fly. Missions could start out simple but end up dynamically changing and become somethingyou regret saying you would do.
Maybe somebody offers you 100cr to safely transport some nobody passenger to Ibootis. Seems simple enough. But somebody interdicts you and asks you to take him to Aulin instead for 10,000cr. He wants him dead though and its an ambush. Or offers you money to eject him into space or turn off life support. This annoys the guy who offered you 100cr and hes paid somebody 100,000cr to kill you.
Decisions, Options, Morally, immorally. Choices you make could have an impact on what happens next.
I imagine Frontiers imagination is better than mine so stick whatever else you can think of in this list to boot too.
One thing is for certain, Its gonna be fun that is for sure.