No single government in the Commonwealth, British or otherwise, exercises power over the others, as in a political union. Rather, the relationship is one of an international organisation through which countries with diverse social, political, and economic backgrounds are regarded as equal in status, and co-operate within a framework of common values and goals. These include the promotion of democracy, human rights, good governance, the rule of law, individual liberty, egalitarianism, free trade, multilateralism, and world peace, and are carried out through multilateral projects and meetings, as well as the quadrennial Commonwealth Games.
The symbol of this free association is Queen Elizabeth II, known for this purpose as Head of the Commonwealth. This position, however, does not imbue her with any political or executive power over any Commonwealth member states; the position is purely symbolic, and it is the Commonwealth Secretary-General who is the chief executive of the organization. Elizabeth II is also the Head of State, separately, of sixteen members of the Commonwealth, collectively called the Commonwealth realms.
Adapted from the Wikipedia.
More information: Commonwealth Home Page.