One important factor to keep in mind is that the various archives, even those with online collections, have a variety of rules regarding access to their collections. Some of the countries make copyright claims to all their government documents and other allow copies to be made without such claims. The National Archives of Australia is a good example of country where many records have been digitized but the country claims copyright to the records. Here is the statement from their website:
The National Archives of Australia supports and encourages the dissemination and exchange of information. All data and other material produced by the National Archives constitutes Commonwealth copyright. The National Archives reserves the right to set out the terms and conditions for the use of such material.
Save for the content on this website supplied by third parties, the National Archives logo and the Commonwealth Coat of Arms, the National Archives has applied the Creative Commons Attribution 3.0 Australia Licence. The National Archives asserts the right to be recognised as author of the original material in the following manner:I strongly suggest that you investigate the extent to which such claims may affect your use of any of the online material available from this and any other archive.
© Commonwealth of Australia (National Archives of Australia) 2013.
Wikipedia has a rather extensive list of national archives. You might be surprised at what is and what is not online from the various archives. For example, the National Archives of the Czech Republic or Národní archiv has digitized all volumes of the Registers of Births, Marriages, and deaths of Jewish communities deposited in the National Archives. In the course of 2014 they have been gradually made accessible on: www.Badatelna.eu. One challenge is that the language used on the website may be outside of those supported by Google Translate although some websites may have a provision to be read in English.
Some of the national archives, such as the National Archives of Estonia or Rahvusarhiiv, have specific resources for genealogists such as digitized, online copies of Lutheran and Orthodox Church records from the parish registers. In this case and in others, you may have to create an account. In contrast, the General State Archives of Greece or Τα Γενικά Αρχεία του Κράτους (ΓΑΚ) has only a very limited online collection.
I suggest that you may find that a trip to the country of your ancestor's origin may include a stop at the country's national archives.