The killing of unborn babies (euphemistically called 'abortion') in Australia remains a subject of state law rather than national law. However, Federal parties and their ministers and policies influence 'abortion' state laws.
The grounds on which 'abortion' is permitted in Australia vary from state to state. In every state 'abortion' is legal to ‘protect the life and health of the woman’, although each state has a different definition. Note: The phrase to 'protect the life and health of the woman' has come to mean 'on request of the woman'.
There is no law anywhere in Australia that requires the notification or consent of a woman's partner.
There is no enforced waiting period for an 'abortion' and a minor does not require parental consent or notification for an abortion - except in Western Australia.
State-by-state (and territory) legality of 'abortion' in Australia
Australian Capital Territory: 'Abortion' law in the Australian Capital Territory was for many years governed by case law and the criminal code of New South Wales. However, in 2002, it became the first jurisdiction in Australia to legalise 'abortion' in full (throughout entire pregnancy), when the Stanhope ALP government, with the assistance of Green and independent members, passed the Crimes (Abolition of Offence of Abortion) Act 2002, removing the killing of unborn children from the criminal statute books altogether; or said plainly, abortion is legal up to birth.
New South Wales: 'Abortion' law in New South Wales is primarily based upon the Levine ruling of 1971 (itself derived from the Victorian Menhennitt ruling of 1969), which declared abortion to be legal if a doctor found "any economic, social or medical ground or reason" that an 'abortion' was required to avoid a "serious danger to the pregnant woman's life or to her physical or mental health" at any point during pregnancy. This was expanded by the Kirby ruling of 1994, which extended the period during which health concerns might be considered from the duration of pregnancy to any period during the woman's life.
An upgraded law passed 8 June 2018 criminalises anyone protesting within 150 metres of an abortion clinic ("protest free zones"). It is deemed an offense to offer pregnant women (and/or accompanying people) within these zones, information related to pregnancy, help and assistance, as well as film, stand or even pray. Protesters face jail time up to 6 months and fine of $5,500 for first offense and 1 year jail and $11,000 fine for subsequent offense.  [1A]
Northern Territory: Legislation in 1974, based on earlier legislation in South Australia and the United Kingdom, legalised 'abortion' in the Northern Territory if the risk to the woman's life or health is greater than it would be if the pregnancy were not terminated and it is likely that the child will be physically or mental handicapped. The 'abortion' must be approved by two medical practitioners and must be performed in a hospital. 'Abortions' must be performed during the first fourteen weeks of pregnancy, except when there is a case of serious risk to the woman's health, when abortions are allowed up to the 23rd week.
South Australia: Medical 'abortion' of pregnancy became legal in South Australia in 1969 under the Criminal Law Consolidation Act, when necessary to protect the life or physical or mental health of the woman - taking into account the current and reasonably foreseeable future - or in cases when the child was likely to be born with serious handicaps. However, the reality is, South Australia now has a situation similar to other states, where women can have an abortion on request.
In SA 'abortions' may be performed by doctors up to 28 weeks. 'Abortions' must be performed in a hospital and be approved by two physicians, and are also subject to a residency requirement. The hospital, dual approval and residency requirement may be waived in an emergency. The term "emergency" is used broadly though, and, according to some staff, has been abused by some doctors.
Queensland: On 17th October, 2018, legislation was passed allowing the killing of unborn babies up to birth. The laws are now similar to what Victoria pushed through in October 2008. 'Abortions' are legal in Queensland up to 22 weeks on request of the mother, after this time pregnant women merely require the farcical "two doctors to agree" policy. It is farcical because one of the doctors is the abortionists and the second a colleague. (Farcical also because South Australia has had the same policy since 1969, and it is well known that a second doctor is not consulted.) The doctor is required to ascertain if an 'abortion' is appropriate, based on the women's current and future physical, psychological and social circumstances. Additionally, a 150 metres bubble zone has been imposed around all the abortion killing centres. Those wanting to give pregnant mothers information, help or assistance can expect to face heavy fines and long jail time. READ ABC ARTICLE HERE
Tasmania: In Tasmania, since 21 November 2013, 'abortions' are allowed on request up to 16 weeks of pregnancy, and then after that time merely requiring the consent of two doctors on medical or psychological grounds - which is incredibly broad. The law also criminalises filming, "intimidation and protests" against patients within 150 metres of abortion clinics ("protest free zones"). See the above NSW definition for "protest free zones". The reality is, 'abortion' is legal for any circumstance, throughout the entire pregnancy - up to birth.
Victoria: Sweeping changes to 'abortion' laws were passed on 10th October, 2008, after Premier John Brumby announced "our existing laws are out of step with community sentiment."  Current legislation now allows abortion up to full term (birth). That is, the killing unborn children is legal to 24 weeks on request, with 'abortions' after that time - up to birth, requiring two doctors to agree that it is appropriate, based on the women's current and future physical, psychological and social circumstances. This was passed by the Upper House 23-17  , with no amendments.
Doctors who conscientiously oppose abortion have to refer women requesting an abortion to doctors who perform abortions or face prosecution. No anaesthetic needs to be given to aborted unborn babies at any stage of development.
Since 2016, it is an offence in Victoria to protest within 150 metres from any abortion killing centre. These are called "protest free zones". See above NSW for definition of "protest free zones".
Western Australia: An 'abortion bill passed, subject to certain amendments, on May 20, 1998, allowing the killing of babies in the womb to be performed up to 20 weeks of pregnancy on request - subject to counselling by a medical practitioner other than the one performing the 'abortion' - or when serious personal, family or social consequences will result to the woman if an 'abortion' is not performed, when the life or physical or mental health of the woman is endangered and when the pregnancy causes serious danger to the woman's mental health. The killing of unborn babies after 20 weeks of pregnancy may only be performed if the child is likely to be born with severe medical problems - which must be confirmed by two independently appointed doctors. In the event of the woman being under 16 years of age, one of her parents must be notified, except where permission has been granted by the Children's Court or the woman does not live with her parents.