If, like me, you live in New South Wales, are of a European background and your family has been here for a number of generations, there is a good chance that your ancestors would have migrated here by ship. If they weren't convicts they may have come as assisted or unassisted immigrants. Assisted immigrants had part or all of their passage paid for them, sometimes by the government, sometimes by a relative and sometimes by an agent (or a combination of all three). Unassisted immigrants paid for their passage themselves. The passenger lists for the ships bringing immigrants to Australia can provide a wealth of information. However, cross your fingers that your ancestors were assisted immigrants because the lists for unassisted immigrants usually only lists their names (often only with an initial) or doesn't even name them at all. Assisted immigrants passenger lists have much more information, and often there are two lists for a ship, one containing more information than the other. The one with more information is the list from the Immigration Board, the other is the Agents immigration list.
The Agents immigration list gives the surname, first name, age, marital status, occupation (calling), native place and county, religion, whether they could read and/or write, and any extra remarks. The Immigration Board list gives all those details plus the names of the immigrants parents, and also the parents place of residence if they are still alive, the details of any relations in the colony (sometimes extremely vague), the immigrant's state of health, whether they had any complaints about the voyage, and remarks by the immigration board. So it is the Immigration Board's list which is most valuable to the family historian.
The NSW Government State records has digital copies of the ships lists online. Use the key name search to find out whether your ancestor is listed in the ships list, ascertain which ship and then view the details in the digital copies. Unfortunately, in my experience, the listing is of the Agents immigration list. To view the Immigration Board lists you need to go to the government records reading rooms, one of the 40 community access points (eg the State Library) or get a subscription to Ancestry.com. It is anticipated that the Immigration Board lists will be added online at some stage.
One other quite useful resource for assisted immigrants is the Immigration Deposit Journals. These record the details of money deposited in the colony to sponsor the passage of an immigrant. Often the money was paid by relatives, to bring out relations, or by someone who wished to employ the immigrant. If the "relatives in the colony" section in the Immigration Board list is vague this may provide a little more detail. Unfortunately it is not available online (update: as of August 2011 it is available through Ancestry.com), but instead in the government records reading rooms or at the community access points.
For more information read the Immigration information on the NSW Government State records website.