Hi. It would be impossible trying to sum up Abbasid ('Abbassi khilafah') achievements/accomplishments in such a short space. However, one could say they had a special role in
(a) development of learning, generally-- in many diverse fields/subjects such as arts, literature, science, astronomy, medicine, mathematics etc; they also built some of the finest and biggest libraries and earliest colleges/universities in the world and in due course, all this knowledge was also transmitted to the West and otjher parts, a true 'global knowledge expansion'.
(b) administration, especially revenue administration and financial management (for example, including banking, budget planning, one of the world's earliest social welfare systems etc); which helped to make the Abbassid empire fiscally strong and secure and helped promote many of their other projects in learning,arts, architecture etc
This is of course just a basic outline. I have also given some basic online links but for a real proper understanding, Id request you please consult books in your local library. One especially detailed account and discussion of this topic is given in Prof Phillip K Hitti's "A History of the Arabs''.