Requirements

Whether you can receive a stay authorization and for how many days depends on the country from which you are:

CountryEntry permit
Afghanistan, Angola, Bangladesh, Burma, Cameroon, North Korea, Ivory Coast, Cuba, Egypt, Eritrea, Ethiopia, Philippines, Gambia, Ghana, Guinea, Haiti, India, Iraq, Iran, Kenya, Libya, Mali, Nepal, Nigeria, Pakistan, Republic of the Congo, Democratic Republic of the Congo, Senegal, Syria, Somalia, Sri Lanka, Venezuela, Vietnam and YemenNot eligible. You can apply for a tourist visa.
Russia180 days
South American countries180 days
All other countries90 days

So if you are from example from the United States or Canada you will receive a stay authorization that is valid for 90 days.

Additional requirements

You must also comply with two other requirements:

1. Your passport is valid for at least six months when you enter Ecuador.

2. You have proof of an onward travel arrangement.
Examples of onward travel arrangements are return tickets and bus or airline tickets to another country than Ecuador. The tickets need to show a travel date within the number of days that you are allowed to stay in Ecuador. Migration officers in Ecuador usually don’t ask for proof of an onward travel arrangement, but airline representatives at airports do it regularly. In such cases, you cannot enter your airplane without proof of an onward travel arrangement. If you cannot show this proof of onward travel arrangement, you won’t be allowed to enter your airplane. When you want to visit Ecuador without a return ticket, you may want to buy a cheap bus ticket to Peru or use a service such as Onward Ticket.

Note: as a visitor, you don't need health insurance. Travel insurance isn't obligatory either when visit Ecuador, but when you travel to the Galápgos Islands you will need to show travel insurance.