Operating Agreement / Bylaws & Minutes
An operating agreement is an agreement among LLC Members governing the LLC's business, and Member's financial and managerial rights and duties. Corporate bylaws are generally concerned with the operation of the corporation, setting out the form, manner or procedure in which a corporation should be run. Bylaws come with minutes of meetings (in the beginning they would include the initial meeting, where bylaws are adopted by the board of directors).
Some banks would require a company representative to present the bank with a copy of those documents (operating agreement for LLC, and bylaws with minutes to corporations) when opening a bank account (albeit its quite rare).
Apostille/Embassy Certification
If you plan to open a bank account in your country or in one of the offshore locations you should be aware of the local regulations and inquire at the bank which documents would be needed to open an account. Typically, at least the Articles of Organization/Incorporation would be required, and this document then needs to be certified for international use.
If you plan to open a bank account in any jurisdiction that is a signatory of a Hague Convention (also called "Apostille Convention") then you would need to have an Apostille attached to a certified copy of your company's Articles (this is done on a state level only).
If you plan to open the bank account in any other country or jurisdiction that is not a signatory of the Hague convention then the process is a bit more complicated and involves certification of the document at the US State Department and then at the Embassy of the country/jurisdiction in question.
In the current banking market, customers can choose between brick-and-mortar banks, typically big national banks with numerous local branches throughout the country, or Internet- only banks.
The advantage of brick-and-mortar banks, such as Chase, Citi Bank, Bank of America, Wells Fargo, or HSBC, is that they have many local branches where you can visit and speak to representatives in person. On the other hand, the main advantage of Internet-only banks, like Silicon Valley Bank, EverBank, ING Direct and HSBC Advance, is that they do not have the overhead expenses of operating local branches, and therefore can afford to offer their customers better rates on their accounts.
Internet-only banks might be more flexible in terms of accepting applications of foreigners, however many of those banks reject applications without Social Security Number.