Answer: Circumnavigation

The explorer you are looking for is Juan Sebastiaìn de Elcano (ca. 1476-1526).

Although Ferdinand Magellan is conventionally credited as the first man to circumnavigate the globe, he died before completing the journey, so the honor can be ceded to the Basque navigator Elcano.

According to..
According to Encyclopedia Brittanica, in 1519 Elcano sailed as master of the Concepción,one of five vessels in Ferdinand Magellan's fleet. On Magellan's death, which occurred in the Philippines in April 1521, Elcano took command of the expedition and brought it safely home to Spain despite scurvy, starvation, and harassment by the Portuguese. Only one ship of the five, the "Vittoria," with 17 other Europeans and 4 Indians aboard, reached Spain in September 1522.

It should be noted that this answer also depends on what you mean by really completed. For example: Magellan himself died mid-journey, but on a previous voyage to Indonesia, had sailed farther east than that point, so he did indeed circle the globe completely (though not in one voyage). Sir Francis Drake was the first to command a successful expedition and survive, returning in 1580 in the Pelican, renamed the Golden Hind, the only ship to survive the journey.

Another perspective
Finally, as Steven Dutch points out on his informative website Circumnavigations of the Globe to 1800, what's at stake is rather a dubious honor, in any case:

"By the 1600's it was possible to go around the world as a paying passenger. Trans-Atlantic trade was firmly established. The Spanish had trade caravans regularly crossing Mexico to link Atlantic and Pacific ports, and they were sending ships regularly between the Philippines and Mexico across the Pacific. Trade between Europe and the Far East was being regularly conducted by several European nations. Thus, there was a continuous network of European trade routes circling the globe. However, it was rarely necessary or useful for a single ship or person to make the complete circuit."

But how do you spell his name?
As to spelling of de Elcano's name, the Library of Congress (LOC) authority record for Elcano notes that the Real Academia de Historia [Spain] declared in 11/23/26 that his name should be written "de Elcano" not "del Cano".

The LOC authority file is the official list which establishes the forms of headings to be used in the LOC catalog for consistency.

Sources

"Elcano, Juan Sebastian de." Encyclopædia Britannica. 2006. 3 Mar 2006.
Entry

Library of Congress Elcano authority record record.
Elcano, Juan Sebastia