I'm going to work on an app that will be consuming and editing data from arcgis server that I have purchased a license for.
I read the licensing documentation but its still not clear to me whether additional licenses for runtime deployment be required in this scenario.
You should only need a Standard license if you're using one of the following functionalities:
License your app
Just to clarify that if your application is using ArcGIS server (or ArcGIS Online) services and works in fully connected environment, you don't need buy licenses. You can find steps how to license Basic level application from the link that Freddie posted.