DLC, or Xenologues , are completely separate missions that can be completed through use of the Dragon's Gate . They will usually reward you with special items, exclusive classes (like Witch), and even some very, very precious skills (like Paragon). You can also get a special character through DLC (Anna).
Xenologues are unrelated to the main campaign in every possible manner. Sometimes they cover the past or an alternate future of the main campaign, and other times they are meant entirely for entertainment, or just to give unique bits and insights into characters who otherwise have little development.