A story mission gradually reveals some of the deeper secrets of why the game's world is in the state it currently is. Story missions usually have cut-scenes at various points, which sometimes give clues to how to further progress the mission. A story mission usually contains fighting elements found in action missions, but can also involve elements like stealth and puzzles found in sabotage and investigation missions. They frequently involve single-player instances.
A single story mission can span over a wide range in difficulty levels, and it should usually not be expected that you can do it from start to finish without also improving your characters skills and abilities.
A sabotage or stealth mission usually, but not always, involves sneaking past enemies and bypassing obstacles rather than fighting them. A sabotage mission can also involve disguising yourself, or hacking into enemy security systems and computers networks.
An investigation mission involves puzzles, thinking and logic. Unlike the other missions you will not always be told exactly what to do and where to do it, but rather have to figure this out on your own based on clues given to you. These clues may or may not be interpreted literally, and some clues may also be red herrings. The clues can often involve searching on the internet or looking up stuff in an encyclopedia. The difficulty level of an investigation mission can range from very easy to insane depending on your prior knowledge and ability to understand the clues.
An item mission is a smaller and fairly short mission. Despite the name it can in reality be either an action, sabotage or investigation mission, and despite the name you will not always get an item reward upon completion.
A PvP mission is a smaller mission you get inside one of the PvP areas in the game. Aside from the fun (or frustration) with PvP the main reward of these missions are PvP-tokens that you can buy gear and equipment with.
When you find a mission in the game world the estimated difficulty level of the mission will be listed after the name. However, this is compared to the mobs in the area(s) involved, and will not always reflect any difficulties that may arise because of stealth, jumping and puzzles.
Currently you are allowed 1 story mission, 1 main mission, 1 dungeon mission, and 3 side missions.
1 story mission at a time isn't really a restriction because you will never have the option to choose a different one.
Most, but not all, action and item missions are repeatable 18 hours after you finish them, possibly giving you an opportunity to choose a different reward.
Some missions also requires you to have completed an other mission before that specific mission can be accepted, creating a timeline in which they are done. Sometimes it will be told which mission you need to do first, but sometimes that field is just blank spaces and you need to figure out this on your own.