A diagnostic is an activity that a teacher carries out with learners in order to identify problem areas they may have. This data is then used to plan further activities.
Before teaching phrasal verbs to a class, the teacher asks the learners to identify all the verbs and their meanings in a prepared text. The teacher observes that most learners know what phrasal verbs look like and how they work, but have difficulty using context to guess meaning. The next class is planned based on this information.
In the class:
Test Teach Test (TTT) is a classroom approach that uses a form of diagnosis, in this case tests, in order to get data on learners and then plan accordingly.