An apostille is a standardised authentication certificate attached to a public document so that it is recognised in another country that has joined the Hague Apostille Convention. Australia is a signatory, and so are Ireland, the United Kingdom, the United States and Canada, which means an apostilled document from any of them is accepted in Australia without the older chain of embassy legalisation.
The issuing authority depends on where the document was issued: the Department of Foreign Affairs in Dublin for Irish documents, the Foreign, Commonwealth and Development Office's legalisation service for UK documents, the Secretary of State of the issuing state (or the US Department of State for federal documents) for American ones, and Global Affairs Canada or a provincial authority for Canadian ones. Apostilles are most often needed for birth and marriage certificates, police clearances, and educational qualifications that an Australian assessing authority or registration body wants to verify.