A senator is an term for a Congressman or Congresswoman. Like the Prime Minister, it is an attempt to tie in the real life political system of Australia into eRepublik game mechanics. "Senators" are considered to be identical to congress members in every aspect except the name, being responsible in Australia for the proposing of laws and the accepting of citizenships.
The Australian Senate (complete list can be viewed here), is responsible for making the laws in Australia. They debate the legislation on the Australian external forums, in the Australian Senate Hall.