IN THE GALÁPAGOS
Large, inhabited islands in the Galápagos will have cell service. However, travelers must activate global roaming with their provider, which can be pricey.
Internet service can be scarce throughout the Galápagos Islands, though some hotels and restaurants may provide wi-fi, and some internet cafés can be found in Puerto Ayora on Santa Cruz and Puerto Bquerizo on San Cristóbal Island.
IN MAINLAND ECUADOR
If you're hoping to stay connected in mainland Ecuador, you're in luck for the most part. Mobile phone use is widespread throughout Ecuador's mainland, though public street phones are disappearing. The majority of nicer hotels offer international and long-distance services, though the surcharges are generally quite high. If you're hoping to make an international connection, using your computer's wi-fi for a video chat or finding your way to an internet cafe could be your best option.
If you're hoping to bring your own phone, you'll need a GSM (Global System for Mobiles) phone activited for global roaming--but watch out for high per-minute charges. It's generally a good option to buy a pre-paid SIM card from one of Ecuador's main cellphone networks.
If you don't want to bring your own phone, it's usually pretty easy to purchase an activated phone with pre-paid minutes on it at a local store or major airport.
Ecuador's main cellphone companies are Porta, Movistar, and Alegro.
WiFi in Ecuador is widespread and growing--just look for the WiFi symbol! It may not always be fast or free, but you can generally find access--especially in the larger towns of Guayaquil or Quito. Internet cafes are a dying breed, however, so it's unwise to solely rely on them for internet access.