If you've played other massively multiplayer games, you know about the drudgery of doing missions, particularly if you've done your second or third character. You go to the quest giving NPC, you click through the same content screens, and go and kill the same monsters every time. You complete missions in a linear sequence, and in the end, the entire process loses a lot of replay value. Think of most MMOGs as having missions like amusement park rides. Like amusement park rides, they can even develop lines as people hang out waiting for a particular monster spawn.
EVE Online missions are different - first of all, they're mostly nonlinear. You can go from nearly any mission to nearly any other mission, and it's not quite as linear as in a traditional MMOG.
Second of all, the missions are completely optional. Unlike a traditional "Dungeons and Dragons" based MMOG, your character's abilities aren't tied to the grind of leveling up. Your character's skills are built on how long you've played and what skills you're trying to improve. While the missions are fun, they're not essential to character growth. It is entirely possible to start a new character, join a corporation, and never do a single mission.
Third, a lot of EVE Online's Missions are built around training you to do specific things in the interface, and to highlight things to do in different parts of the virtual world. While completing missions will get you gear and money, there content of the game revolves less around quests (and "patrol grinding") than it does around engine building and preserving your access to Eve Isk.
Most EVE Online missions are built on one of five different levels, depending on the agent you're getting them from. Level 1 missions are the easiest, and can be handled with a frigate (upgrade to one as soon as you can). A destroyer makes most of the level one EVE Online missions into boring cake walks. Level 2 EVE Online missions are against more numerous opponents, and can put you up against a cruiser - make sure you're prepared. At level 3, EVE Online missions are never cakewalks, and often start with you at a significant disadvantage. At level 4, you're expected to work with a team, and some level 5 EVE Online missions use capital ships, and are so difficult as to be considered suicidally tough Eve Online Isk.
One of the hidden benefits of doing EVE Online missions are loyalty points. While your skills don't change as a result of running missions, your cash flow does, and with loyalty points, you can get access to gear you otherwise wouldn't get.
