I want to use ArcGIS Online in public web application
or limit users or any pricing?
For example I know that
Any information will be appreciated
It can be free! You get 50 credits per month. http://www.esri.com/software/arcgis/arcgisonline/credits
It really depends how you would like to use it. For more information regarding pricing - https://developers.arcgis.com/pricing/