It all depends on what country you're from as to what arrangements Australia will have for visas. Being British, it was very easy for me to apply online. There's a list of eligible countries on the Australian immigration website. There are some nationalities that can't get a Working Holiday visa, but they can apply for a Work and Holiday visa instead, although only US citizens can apply for this one online.
It was a simple application form for my Working Holiday visa, available on the immigration website, and it took me just over an hour to complete it. Within 24 hours of submitting it I had an email telling me that my application had been approved and my payment had gone through. It cost me AU$420. And that was it! Really easy and painless.
There are certain requirements you need to meet to apply for the visa.
You need to be between 18 and not yet 31 years of age, you need to be outside of Australia at the time of application, and you can't travel with any dependent children.
You can apply any time within the 12 months prior to your arrival in Australia, but you have to enter the country within 12 months of it being granted.
Your visa is automatically activated when you arrive in Australia, so it feels like you haven't done anything, but you can check that the visa is active on the Visa Entitlement Verification Online (VEVO) website.
And then you've got a year to work and travel around Australia! You can leave and return as many times as you want during that year without applying for a re-entry permit like you need to with some countries. You can only work for one employer for a maximum of six months though, which can be a deal-breaker when job-hunting.
If you want to apply for a second year working holiday visa then you'll need to do 88 days of rural work (it doesn't have to be with the same employer or consecutive days), and be able to prove that you've done it. Lots of people will sign you off for your second year visa even if you haven't done the exact type of work they are looking for. But if you do go for something like this, do be aware that it is illegal, and you need to trust that they will actually sign you off for it at the end of the three months you work for them. Also check that their post code is eligible to be classed as "rural work" as that's another thing that dictates whether or not you're awarded the second year. If you are granted the extra year then you can either stay on and do the year straight away, or leave and come back before you turn 31.