The question asks for the name of the political movement that advocate Hinduism as a basis for society or state.
Society or state?
I go with state first.
No such words exist actually in the present. However, let's call a group of people called ASDFGHJKL. They want Hindu Rastra. Hindu Rastra means "Hindu country". We don't actually have a word to call these movement. Maybe we can call them "Advocates of Hindu Rastra". They advocate a radical form of society.
Hindutva as pointed out in the other answer is a misnomer. It means Hinduness. There is no Litmus test for Hindutva. Maybe there exists a pH scale for Hindutva. A devout Hindu is a worshipper of his land i.e his country because Lord Ram (a God of the Hindus) worshipped his land. He might not fall into the category of the "advocates of Hindu Rastra".
To simplify, "Advocates of Hindu Rastra" is a subset of "Hindutva believers". I safely say if I were to draw the Venn's diagram, the ratio of the subset and the superset would be very big. I leave you with the liberty of the calculation.
Society
Most Hindus live in India. There are countries where Hindus are in vast numbers. I will explain about India.
Society is diverse in India. That is one thing, it's important for OP as well anyone not from India needs to understand.
Now every village has their own customs. These are sometimes reflected in their lives religiously. People of a place I stayed some time while ate vegetarian food on Monday. People of another place just 10-15 kilometres aways do it on Thursday. I mean most of them do. It is based on some religious belief they hold dear to. It is not just about food. I am from a cosmopolitan place. Some families in one particular place pray every evening. They sing it aloud. Others don't. To be honest, there are villages resided by these people. It is not for Hindus only. When I studied in college, wine shops were shut on Sundays in the city where I stayed. I am a Christian and studied in a Christian College. Most people are Christians there. Now what can someone call these diverse groups of people? We just have words to describe them.