For the instance of
render :json => @projects, :include => :tasks
You are stating that you want to render @projects
as JSON, and include the association tasks
on the Project model in the exported data.
For the instance of
render :json => @projects, :callback => 'updateRecordDisplay'
You are stating that you want to render @projects
as JSON, and wrap that data in a javascript call that will render somewhat like:
updateRecordDisplay({'projects' => []})
This allows the data to be sent to the parent window and bypass cross-site forgery issues.