If you think that the definition of who a refugee is which has been explained here fits your situation, then you may be able to apply for a Protection visa to stay in Australia. A Protection visa is a permanent visa which is granted to person if they need protection in Australia, because their situation meets the definition of who a refugee is. To find out more about what a Protection visa is and how to apply for one please click here.
If you arrived by boat you will be in a slightly different situation to the one described in this website. If you arrived by boat, and you want seek asylum in Australia, please click here for more information.