I'm looking for a list of all the banks and their details, or just the biggest ones, by country. Is there any?
Wikipedia has lists of all the banks in certain countries, those seem pretty accurate for the larger banks and countries with more active wikipedia communities.
For more in depth data, I think you might have to get the data for each country seperately through their financial supervision department. The United States has that data here: https://cdr.ffiec.gov/public/
A good place to start for the EU is here: http://www.efsf.europa.eu/about/links/index.htm
The Dutch association for banks has a list of members here: https://www.nvb.nl/vereniging/120/leden.html
Furthermore, companies like pwc publish reports on the banking sector, such as this one for China: http://www.pwccn.com/webmedia/doc/634442705425169010_fs_foreign_banks_china_jun2011.pdf