Australia is a very large country, so it really depends on where you plan to go and what you want to do.
If good weather is you primary concern Spring and Autumn are ideal for visiting most parts of the country, with pleasant, warm and sunny conditions for the most part. Personally I prefer October/November or March/April, which is a little closer to summer than in May or September as suggested by Jangaplanet.
The Australian summer, which is December through February, can be very hot and is not the best time to visit the north of Australia (including the Great Barrier Reef and Kakadu)... this is monsoon season in the tropics with hot/humid conditions and the occasional cyclone thrown in. South of the tropics you can expect scorching dry heat up to mid-40s Celsius (110+ Fahrenheit) in many places including the state capitols of Melbourne, Perth, and Adelaide, with the outback being even more extreme! Sydney is also very hot in summer but tends to be wetter.
Winter (June through August) can be a good time to visit northern Australia and the outback due to the more tolerable, tamed weather!