If your ancestors immigrated to the United States, you'll have the ability to search by passenger lists (the ship's manifest), citizenship and naturalization records, border crossings and passports, and even pictures and descriptions of the ships that transported them to America.
Your ancestor's surname may have been recorded wrong by immigration officials/ be different than what you know it as. Also, your ancestor(s) may also have "Americanized" their first name after being in the United States, they would not have wanted to "stand out" with a name that clearly marked them as an immigrant. So immigration records may show Pavlo (the Slavic version of Paul) but later on an application for a marriage license will show the name Paul listed instead.
When searching the passenger lists, try to enter in as much information as possible even if it might be a guess. You can also click on the "exact" button and this will then allow you to list an approximate year give or take -x number of years (as you can see from the screenshot below).
It may be very difficult to decipher the original document. But the source citation will list the line number for your ancestor(s) making it easier to read the manifest list. This is the document that the ship's captain would have presented to immigration officials at Ellis Island. It also lists the date that the ship left the European port (in this case it left Antwerp, Belgium on October 11, 1913).
The ship's manifest will contain information on your ancestor like his ethnicity/nationality, martial status, age, birth place, departure port (if emigrating from Europe, what port in Europe they left from), arrival date (the date they arrived in the United States or in some cases, Canada), arrival port (what port in the United States they arrived at), final destination (where they were ultimately going to and settling in the United States), physical features (i.e. height, hair color, eye color, complexion), money in possession (immigration officials required all immigrants to have a certain amount of money on them when they arrived in the United States to show they wouldn't be a financial burden on the American government), and the name of the ship they traveled on.
*Castle Garden was the name of the immigrant inspection and processing station that existed before Ellis Island was built. It was also located in New York harbor.*
Depending on when your immigrant ancestor came to the United States, if it was before 1918 and they came from what was then the former Austro-Hungarian Empire, immigration officials would have just labeled them "Austrians" even if their actual nationality was Hungarian...Czech...Polish...Ukrainian...Croatian, etc. Many ship manifests/immigration and naturalization records incorrectly list the nationality of immigrants (a combination of both borders/empires that existed at the time and also immigration officials not caring to record the correct nationality.
You may also notice words are misspelled or have various misspellings of the same word.