The new law changes the definition of marriage from "the union of a man and a woman" to "the union of 2 people" [text, PDF]. This comes after years of resistance from conservatives. There had previously been 22 unsuccessful attempts [NYT report] to make this change since 2004.
This particular vote was preceded [Sydney Morning Herald report] by a national postal survey in which 61% of over 12 million respondents were shown to be in favor of marriage equality. The response prompted the government to allow a free vote in Parliament.
The vote caused immediate celebrations, including cheers and even a song, with supporters singing "I am, you are, we are Australian." Personal stories were shared by numerous senators and MPs, including one that played out on the floor of parliament as Tim Wilson [official website], an MP from Victoria, proposed to his partner [video] during debate on the bill.