An agile retrospective, or sprint retrospective as Scrum calls it, is a practice used by teams to reflect on their way of working, and to continuously become better in what they do.
The whole team attends the retrospective meeting, where they ?inspect? how the iteration (sprint) has been done, and decide what and how they want to ?adapt? their processes to improve. The actions coming out of a retrospective are communicated and done in the next iteration. That makes retrospectives an effective way to do short cycled improvement.
The retrospective facilitator (often the Scrum master) should have a toolbox of retrospective techniques, and be able to pick the most effective one. Some of the techniques to do retrospectives are asking questions, state your feelings with 1 word, 5 times why (Root Causes) or asking why, Agile Self-assessment Game, solution focused/strengths and retrospective of retrospectives.
WHY to take retrospectives
So if you want to solve the problems that you are having, and deliver more value to your customers, you have to change the way you do your work. That is why agile promotes the usage of retrospectives: To help teams to solve problems and improve themselves!