The Minister has powers under the Migration Act 1958 to intervene in your case when the Minister thinks it is in the public interest to do so. Before you can seek intervention you must has had a decision from the Department of Immigration and Border Protection and have reviewed that decision in the Administrative Appeals Tribunal.
What is and what is not in the public interest is for the Minister to decide. The Minister is not legally bound to intervene or to consider intervening. When the Minister intervenes to make a more favourable decision, this usually means that the Minister grants a visa. However, only a small number of all requests for ministerial intervention are successful.
It is important to note that there are strict time limits for making an application for Ministerial Intervention. Generally a Bridging Visa will be available to you whilst you await your matter to be considered.