The word subsidiarity is derived from the Latin word subsidiarius and has its origins in Catholic social teaching.
The principle of subsidiarity holds that a larger and greater body should not exercise functions which can be carried out efficiently by one smaller and lesser, but rather the former should support the latter and help to coordinate its activity with the activities of the whole community.

This principldefines subsidiarity as the idea that a central authority should have a subsidiary function, performing only those tasks which cannot be performed effectively at a more immediate or local level. It was introduced to the European Union in the Treaty of Maastricht as a general principla applicable to all areas of non-exclusive competence.

"The principle of subsidiarity holds that government should undertake only those initiatives which exceed the capacity of individuals or private groups acting independently. The principle is based upon the autonomy and dignity of the human individual, and holds that all other forms of society, from the family to the state and the international order, should be in the service of the human person. Subsidiarity assumes that these human persons are by their nature social beings, and emphasizes the importance of small and intermediate-sized communities or institutions, like the family, the church, and voluntary associations, as mediating structures which empower individual action and link the individual to society as a whole. "Positive subsidiarity", which is the ethical imperative for communal, institutional or governmental action to create the social conditions necessary to the full development of the individual, such as the right to work, decent housing, health care, etc., is another important aspect of the subsidiarity principle."[ Taken from Wikipedia ]

In other words, it means the State shall take action only if and insofar as the objectives of the proposed action cannot be sufficiently achieved by the communities -Society- and can therefore, by reason of the scale or effect of the proposed action, be better achieved at the State level.  It means that policies should always be made at the lowest possible level, and that the higher level should only legislate when there is unanimous agreement that uniform regulation is necessary.