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When were they born? Civil records in present-day Hungary are nearly complete for those born 1895 and after.

I'm using the familysearch,org website (from the Mormons*) family tree programme. It's quite OK and above all else, it's entirely free. 

I don't know if it has any Hungarian document uploads but it covers the USA, Canada, UK etc quite comprehensively, so it might have some linked data to other records to give a bit of a leg up.  I know it cannot reach German records - not that I've found anyway. Maybe it won't get as far as Hungary.

*I'm not a Mormon, just somewhat of a freeloader.

@zif they were born in the 1860s and 1870s unfortunately

@fluffy2560 thanks! That's where I found the census documents. It doesn't seem like they have Hungarian documents, though

-- Church records are official birth records in Hungary before 1895. Family Search has most of these. Look for church baptism records.

-- You don't say where in Hungary your ancestor was born. Prior to 1918, "Hungary" was much larger than present-day Hungary. If your ancestor wasn't born in present-day Hungary you need to search in the present-day country where he was born.

-- Marriage records are important not just to show lineage but legitimacy.

-- I'm exceptionally doubtful you'll get far without a Hungarian starting document. If you can't get a birth record, perhaps an old Hungarian passport might do, I don't know. But you need something from Hungary (as it then existed) to start your lineage.

-- Remember that for Simplified Naturalization you must have reasonable conversational Hungarian.

    @fluffy2560 thanks! That's where I found the census documents. It doesn't seem like they have Hungarian documents, though
   

    -@jgreenspan02

You might be able to nail down other information like dates of birth, parents names, village names, original names etc.

I was just today trying to find my ancestor born early 1800s in Hamburg, Germany. 

I think I'll have to ask the government there but it was Prussia back then. 

They only seem to have records from late 1800s.