CBT came first. It was first developed by Aaron Beck and Albert Ellis in I believe the 1960's. It's premise is that negative feelings and unhealthy behaviors are related to irrational thoughts. It's kind of like a triangle, where each of those--thoughts, feelings, behaviors influence each other. The theory is that changing the thoughts tends to be the quickest way to change the other two.

DBT takes some elements of CBT and adds to it mindfulness and distress tolerance skills. DBT tends to take longer and be more intense. You might go to group therapy and individual therapy for example. But it has shown some promise in helping people with borderline personality disorder, which tends to be pretty complicated.