Obtaining your residency in Ecuador requires you to do your "Due Diligence" and make sure that you have plenty of patience. The information below may be current at the time of writing this but is subject to change at any time. For more information you would need to contact the Ministry of Foreign Relations. But this should give you some of idea of what to expect.
Some expats have mentioned on various blogs that you do not require an Attorney to help you with your paperwork. My advice is that if you do not speak fluent Spanish and have an understanding of the legalities required to be adhered to in Ecuador - "Get an Attorney".
Ok so what happens when you land in Ecuador. Quite simply you would normally arrive on a Tourist visa and get a T-3 stamp placed in your passport. This allows you up to 90 days in the country. If you are planning on staying longer than 90 days then seek advice from your Ecuadorian Consulate. Here you can apply for a 12-IX visa BUT this can also be done while you are in Ecuador. Just don't leave it to late as this things take time in Ecuador.
You actually need the 12-IX Visa before you can apply for residency in Ecuador. It is a requirement that needs to be filed along with your residency papers. So what do you do if you arrive and decide you want to stay longer and apply for the 12-IX visa. Find an Attorney to help you out. If you decide to go it alone then I wish you good luck.
A resident visa allows you permanent residence and you can still travel outside of Ecuador BUT yes another "but", you cannot leave the country for more than 90 days per year in the first two years of your residency.
There are several different visas that you can obtain and the one of the most used by retirees is the 9-I Pensioner Visa. Pensioner Visa Requirements
If you are considering investing at least $25,000 for one person then you can obtain a 9-II Investor of Real Estate Visa. Now this can be monies put into a CD or property.
The following links may help you out a little further.
Consulates and Embassies in Ecuador
Ecuador Consulates in the USA
Travel.State.Gov Us Department of State Visa and travel information.