In genealogy Immigration Records refer to information about individuals who moved from one country to another country where they intended to become citizens or at least become workers and/or residents (permanent or temporary). So any information that deals with departure, travel, arrival and citizen or worker registration for individuals can be considered immigration records.
Important genealogy information, like date and country of birth, names of father and mother, date of immigration, occupation and more, can be found in some immigration records.
Most early immigration to the United States was by ship and so ship registries or ship passenger arrival records are an excellent place to look for immigration records. You might be able to find an individual's age, place of birth, place of last residence and more. You might also be able to determine an individual's traveling companions if any.
In many cases you can find an individual's year of immigration to the United States and Naturalization information in the Federal Census records. The year of immigration and whether or not an individual was Naturalized were two questions in the 1900, 1910, 1920 and 1930 Censuses.
Naturalization is the process by which an alien (not born in the United States) becomes an American citizen. This process took place in the Courts and depending on the year it could have occurred in a Federal, State or Local Court. The National Archives Immigration Records is a good source of immigration information but few of its records are online.
Individual personal records are another good source of immigration information. Look for passports and other documents that might contain information about departures and arrivals.
Two excellent websites where you can find some Immigration Records online are CastleGarden.org (a searchable database of 11 million immigrants who arrived in the US between 1820 and 1892) and Ellis Island (a searchable database of information about passengers and ships that entered the Port of New York between 1892 and 1924).
For Naturalization information you should visit the websites for the local (County or City) genealogy society and historical society. Here you can probably find the court where the Naturalization process occurred and where the information can be obtained. There may also be indexes that can be very helpful in your search.
If you are looking for the country of birth of an immigrant to the United States try looking in the US Census. If you can't find the immigrant in the US Census look for his or her children in the US Census.
You can find the father's place of birth and the mother's place of birth in many census years - 1880, 1900, 1910 and 1930.