Missions are what makes up the majority of gameplay in DCG. There are a few different kinds of missions but the jist of it is to complete a series of tasks within a specified amount of time. A succesfully completed mission with result in receiving a special reward item, usually a clothing item.
Crafting Missions are the most common mission in the game.
Gifting Missions gimmie things
The Travel Journal missions are a special kind of crafting mission that take place over a longer span of time, usually around a month. So far there have only been two travel journal missions: summer 2013's Venice Vacation and Halloweens Spooky Cafe 2013.
Here is the Time Line with Crafting Missions, Gifting Missions and Collections.