[Vuejs]-Name the Rules and Demands that define When Render function must Run?

I think you are looking for this: Reactivity in Depth – https://v2.vuejs.org/v2/guide/reactivity.html

If you are more of a visual person… then the lifecycle diagram might be easier to understand – https://v2.vuejs.org/v2/guide/instance.html#Lifecycle-Diagram

Basically any time a reactive data property changes… Vue will re-render the component.