Does an EU citizen need a visa to visit Australia for holidays?
Yes, a visa is required for EU citizens.
Fortunately, getting one is free and (usually) pretty quick. The easiest way is to apply for eVisitor online. See "Eligible passport holders" on that page to check if you are eligible for eVisitor.
The eVisitor allows visitors to travel to Australia for short term business or tourism purposes for up to three months. eVisitor applications are free and are available to passport holders from the European Union and a number of other European countries.
More information on Australian visas can be found at https://www.homeaffairs.gov.au/trav/visi
Personally this surprised me last spring (as pampered EU/Nordic tourists rarely need a visa and I assumed a passport would suffice). I'm glad someone happened to ask me about it while I still had a week or two in my previous destination before flight to Australia.
If you are going to some of the small island such as Norfolk Island you should do some reading as they come under different rules. http://wikitravel.org/en/Norfolk_Island#Get_in
Interestingly, Australia changed the name of the thing to pretend it's not a visa but a mere electronic authorization like the US ESTA for the sake of reciprocity. It is easier to get than a regular visa but refusal rates for some EU countries like Romania and Bulgaria are close to 20%, which is why they consider it to be a visa and asked the EU Commission to take action.