If you are lucky with your census search, you may discover the date your ancestor immigrated and if he and/or she was naturalized. Armed with these discoveries, you can then look at available passenger records. Passenger records list the name, age, and sometimes place of origin of the immigrant ancestor as well as any family members. It also lists places of departure and destination, ship name, date of arrival. Naturalization records may contain more detailed information about the ancestor's homeland including town or city, county, and country as well as an exact birthdate.
Armed with such information, passenger lists and naturalization records, serve as a gateway to your ancestry beyond borders and ultimately a look into the many cultures that have contributed to make your family what it is today.