If you're going to move or just visit Ecuador the first thing you need to get is a Passport.
If you do not have a Passport you can go to www.travel.state.gov/passport . This is the US department of state. All the forms are available from this site.
If you need to get a passport fast, there is an expedient one and one that you can get quicker (like in 3 days). But you have to have your tickets within 14 days of leaving on the plane. It will cost you more money to do it, but if you do need to get one ASAP, it can be done.
There are private party companies other then the site above to help you expedite a passport fast. Getting a passport is usually simple but they are getting harder to get nowadays.
In my opinion, they are closing borders. There are already over 2 million people on the No Fly List as of the date today (Aug. 18, 2009). I believe as the days go by, it will get harder and harder to get out.
So, get your passport! It doesn't matter if your just visiting or moving, get your passport and you're good to go. Let me know if you have any questions.