The Transferable Preferential Ballot (TPB) (sometimes incorrectly called the "Australian ballot") is used in all WSFS voting: for the Hugo Awards, Worldcon Site Selection, and various internal elections such as for the Mark Protection Committee. TPB is designed to make it more likely that widely-acceptable candidates will win over candidates with a smaller but more zealous following.
Each voter ranks the choices from their most-preferred to their least-preferred, numbering them from 1 to whatever.
When the ballots are counted, first-places votes for each candidate are counted. If a candidate received a majority of votes cast, that candidate is the winner. If no candidates gets a majority (which is usually the case for the Hugo Awards, since there are normally five nominees in each category), the candidate with the fewest votes is eliminated. The ballots which listed that candidate first are redistributed to each voter's second choice. (Ballots with no second choice are discarded.) If a candidate now has a majority of the ballots remaining, that candidate is the winner.
If there is still no winner, the same procedure is followed again: the candidate with the fewest votes is eliminated and that candidate's ballots are distributed among the remaining candidates or discarded according to the voter's preference, and the remaining candidates are again checked for a majority winner. This cycle continues until there is a majority winner.
The effect of this is that, unlike the more traditional voting systems, if your preferred candidate is eliminated, your preferences among the remaining candidates are still taken into account. A candidate with narrow but enthusiastic support will not win even though that candidate may have the most first-place votes because the larger group of voters who do not like that candidate will not have selected it as a second (or third, or fourth) choice and thus that candidate's vote total will not grow during the elimination rounds.
The process for determining second place is not to look for who is eliminated last, but to start over with the ballots which voted for the first-place winner redistributed and then following the same process to determine second place. And so forth for third and later places.
This is all standard practice. WSFS has one special additional check in the Hugos: Once the winner is determined, the Hugo Administrator counts the voters who preferred No Award to the winner, and if a majority of voters preferred No Award over the winner, the win is disqualified. (This has never happened, but is in place to prevent a winner from being declared when the electorate is so badly split that most voters would prefer no award to be given than one or more of the choice(s).)