You can find work teaching English in Ecuador, despite the red tape. A great many English teachers, and travelers, enjoy spending time in this small equatorial country. There is both paid work teaching English and voluntary work available. Some of the voluntary schemes charge, but this is not necessary, if you don't mind arranging some things yourself.
If you plan is teaching English in Ecuador for money, you are better to stick to the main urban areas of Quito - the capital - Cuenca or Guayaquil, although Cuenca may be more difficult. If you want to teach in a jungle village or a small town, you will probably need to to look at opportunities for voluntary work. There are obviously exceptions to this, there is a limited amount of paying work in the provinces and some voluntary work in the cities.
If you are interested in voluntary work teaching English in Ecuador look on the notice boards in the hostels, cafes and other locations in Mariscal, the traveler's center of Quito. Also take a look at this site for volunteers Volunteers South America If you are interested in being paid to teach English you should still take a look at the notice boards in the tourist area, while enjoying your big macs there.
You could try the classified sections of the Ecuadorian newspapers online, although the times I've looked there has been nothing there. An internet search for newspapers in Ecuador will give you these sites.
Also do an internet search for English language schools in Ecuador. This will return some lists of language schools, which you can approach from outside the country by sending your resume and a cover letter. Call them again when you arrive. The LA Job List has such a list of English schools in Ecuador. Probably the best places to look for work from outside the country, apart from googling for lists of schools and applying directly are Dave's ESL Cafe and TEFL.com.
Tourist visas are not extendable. If you want to teach English in Ecuador, get a 12-IX visa from your nearest embassy before you arrive. Working on a tourist visa is done by some, but is illegal, and, even worse, you will probably end up working for the most dodgy language school owners, with all the problems that entails. A 12-IX visa will allow you to teach English for 6 months. It is not renewable. If you want to stay longer, make sure you have a degree and TEFL certification, and find an employer who will arrange a work visa for you. The 12-IX visa can be converted into a work visa. You will need to make a one year commitment.
To get a 12-IX visa, first take a deep breath - there's a lot of red tape involved. You will be asked to produce many things. Take everything you can think of, and then some more. Requirements differ, depending on which country you are applying from, so check with you local Ecuadorian embassy before you go, but here is a list of things which are likely and have been asked for before.
Bank statements, passport, return tickets, travel insurance, photos, health check (possibly including an HIV test), criminal background check, job offer letter (not needed if you have enough money in the bank), visa application, visa certificate (these two can be printed from the US embassy website) and some people have even been asked to produce their marriage certificate.
When you arrive in Ecuador, you will then need to register - more bureaucracy - then you will need to get a censo [in the same office in which you register]. Take every document and piece of official paper you have with you, and expect some waiting. Things you may need are color photocopies of your passport ID page, visa page, entry stamp page, registration stamp page, proof of where you are staying, a little cash, color passport photos, a manilla envelope. Expect to have to deal with staff who only speak Spanish - and good luck.
If you only have a few months, and just want to hang out, I'd recommend forgetting teaching - if you don't really need the money - and taking a Spanish course, of which there are many here.
While most people are fine, there is some problem with crime in Ecuador, especially in the bigger cities. Guayaquil has many reports of the dangers of taking taxis, for example. Most crime is petty crime, but some is not. Take care in taxis (don't flag them down in the street), and at night. Also, remember that if you take care, and don't flash expensive items or money around, you will probably be alright.
A new centre offering CELTA in the Galapagos Islands has opened. There's a thread in the forum below discussing it.
For more information on teaching English in Ecuador, I recommend asking some questions at Dave's ESL Cafe, on the General Latin American forum