Question
How do I propagate a peony plant and after bloom there is a horny thing  where the flower was ,what is it?

Answer
Hi Caroline,
Thanx for your question.  The best method for propigating peonies is to separate the rhizomes after blooming. Each fleshy root division should have at least 3 to 5 structures called eyes. The eyes look  like pink buds and are the shoots for next year's growth.

Plant the eyes about one to two inches deep. If one plants too deeply the plant will produce all foliage and no bloom.  In colder climates, you can put a good layer of mulch around the plant.  

The horny thing you describe where the flower used to be is a seed pod structure but I find that all of mine have immature, underdeveloped seed.  If you ever do find viable seed, you can try to germinate but it takes almost 3 years for germination and the seed should be exposed to constant warm/cold alterations in climate to break a very deep dormancy.  To me it's not worth the time and trouble unless you are a hybridizer or scientist.  I hope this helps.
Tom