Hi Brian,
Right now you can come to Ecuador on a tourist visa and you will automatically be allowed 90 days in the country. After you arrive you can apply for a 12-IX visa which allows you an additional 180 days from the time of issue. You can also apply for this at your local consulate in North America before coming down if you wish.
The immigration laws and visa requirements are always a moving target here and in fact a new law has been passed (but not yet put into effect) that may change what is required. To obtain the 180-day and/or a permanent visa, at a minimum you can count on needing an apostilled birth certificate, an FBI background report, a valid passport, two passport-sized photos, and all applicable forms.
Your best option, especially with potential changes coming, is to contact your nearest Ecuadorean consulate or an immigration attorney to be sure you are in compliance with the most current requirements.
.
