Not only do FDIC and NCUA provide protection for depositors against bank/credit union failures, but also ensure a level of standards. Being a member of FDIC or NCUA ensures that the institution meets a long list of regulations (see FDIC regulations and NCUA regulations). Many of these regulations are designed to protect customers. Foreign banks may be legitimate, but it's hard to know what standards they are required to follow.
For most of my posts, I include a link to the FDIC or NCUA page which shows the government information about the institution. One important piece of information is the URL of the institution. This can help verify that the website really belongs to the institution. Just click on my FDIC or NCUA link on the bottom of the posts.
If you're trying to determine if a bank is FDIC insured, go to the FDIC's Bank Find Page.
For credit unions, go to the NCUA's Find a Credit Union Page.
A few credit unions are not insured by the NCUA. All of these that I'm aware of have private insurance by ASI. These are legitimate credit unions. You can verify that a credit union is ASI insured from the ASI's Credit Union Search Page.
[Edit 1/31/08: Updated ASI info. Patelco Credit Union is no longer privately insured by the ASI. It's now federally insured by the NCUA.]
[Edit 2/08/09: Updated URL to ASI search page.]
[Edit 5/28/09: Updated URLs to NCUA pages.]
[Edit 12/1/11: Updated NCUA URLs]