As for the actual website you mentioned, that is a particular ministry (!) of a particular man, who is trying to organise conferences and seminars for various people (especially business people) to learn how they can influence their surrounding circles and contacts.
The website holds to the "seven mountains" concept which is nothing other than an informational template for understanding influences in social systems and societies. The template divides society into seven spheres (called "mountains"), and posits that whoever is at the top of those spheres guides the society. If the ungodly occupy the top positions of influence in those spheres, then the society proceeds in an ungodly direction. If however the godly occupy the top positions of influence in those spheres, then society proceeds in a more godly direction. It is based on an analysis of historical societies for the past 2000 plus years.
The "mandate" is understood as a call for Christians to use their position in society to further influence the people around them, and hopefully to get the top positions in each sphere occupied by Christians. It should be noted that this mandate does not involve "legislated influence" but has to do with "winning friends and influencing people" through both personal evangelism and lifestyle witnessing.
http://7culturalmountains.org/pages.asp?pageid=107994
Q: Is the Reclaiming the 7 Mountains strategy about Christians taking over?
A: No, it is about encouraging Christians to serve the culture by solving societal problems through love, compassion and service as following the example of Jesus Christ. It is about serving those in key cultural spheres that the Church has abandoned. The original mandate in Genesis 1 calls for Adam and Even to exercise stewardship over the earth as God's representative. However, when Adam and Eve fell this mandate could only be restored through the death of Jesus (Luke 19:10, Col. 1:19). We do not believe Christians are to control the world or seek to have a utopia in society. However, we do believe we are called to serve the culture as we live out our faith to be salt and light in all aspects of society.