What is a research assistant in the Australian system? I've noticed postings for positions with the title “research assistant” from Australian universities. They require a PhD and are often (or exclusively) one-year contracts. Are these positions akin to anything in the US system? Are they more of a trainee position (like a postdoc)? Or is it more of a senior position within an established lab (like a research assistant professor)? Is it something else?

The one-year contracts suggest that they might be soft money positions. Are they typically renewed or transformed into more permanent positions?
Job advertisements in Australia invariably list a level.  If it says "Level A Academic" it is a postdoc.  If it says "HEW Level Something" then it is not a postdoc.  Consult the unversity's Enterprise Agreement for detailed definitions of the levels.  An enterprise agreement is the same as a collective bargaining agreement.  Note that starting pay for Level A is usually much higher if you hold a PhD.

http://www.nteu.org.au/rights/agreements