The first of the four mission types is known as a rank mission. Rank missions comprise the main campaign of Spiral Knights and will allow players to climb the ranks of the organization. Each rank mission will grant players pre-set awards in addition to any loot found on the mission itself. Prestige missions, the second type of mission, are daily tasks that will reward players with prestige. Players with high prestige scores will receive an unspecified bonus sometime in the future. The third mission type, expansion missions, operate similarly to rank missions in that they "feature set rewards," but the missions must first be "unlocked with purchase," though as of now we don't know whether that means a microtransaction or an in-game-currency purchase. Lastly, we have arcade missions, which are the same arcade missions that players know and love, except now they can be joined directly from the mission interface instead of making you run to the arcade itself.
For the full details on all the new mission types and features being added in the new patch, just head on over to the Spiral Knights forums and take a look.