What famous string of islands are about 600 miles west of mainland ecuador?

The famous string of islands that are located about 600 miles west of mainland Ecuador are the Galapagos Islands.

Explanation:

The Galapagos Islands is an archipelago belonging to Ecuador. The 19 islands are located in the Pacific Ocean about 1,000 kilometers west of mainland Ecuador. The archipelago comprises 18 major islands and about 100 smaller islands. The archipelago is also the province of Galapagos as a separate province among Ecuador's provinces. The equator goes over the largest island, Isabela. The Galapagos Islands are known for the large number of endemic species found here and also for Charles Darwin's research that resulted in his theory of natural selection.

Galapagos Islands