Sverige is the Swedish term for Sweden. It comes from Svea-rike, which means Land of the Svear, Swedes, or Svenonians, as Tacitus has it. The Svear were an expansionist people who originally inhabited the part of Sweden called Uppland. Through colonising and mixing with other people, they would give name to the entire country. It was the Earl Birger who decided this when he founded Stockholm, the Swedish capital.
The other two main peoples of the country - excepting the Sami, who did not count at the time - were the Gutar and the Götar. The people of these tribes were gradually assimilated into the main Swedish culture. Their names, however, can still be spotted in the island Gotland and the city of Göteborg.
"Sverige" is generally pronounced without the G, so something like Sver-ye. Similarly, the country of Norway is called Norge in Norwegian. The Swedes tend to drop the G here as well.