Hawaii was an independent, sovereign kingdom from 1795 until its monarchy was overthrown by American plantation owners with the support of U.S. Marines in 1893.
Texas became an independent republic, known as The Republic of Texas, in 1836, when American settlers overthrew the Mexican government. It existed until Texas was was admitted to the union in 1845. Only a handful of countries ever recognized Texan independence, though.
Vermont also claims to have been an independent country, known as the Vermont Republic, from 1777 to 1791. Virtually no one recognized Vermont's independence, however—not even the Continental Congress.
The Iroquois Confederacy has been a sovereign political unit for at least 800 years. A 1791 treaty with the U.S. government recognized that sovereignty. While the Iroquois (along with 566 other recognized tribes) are not independent countries, they are considered sovereign under the U.S. Constitution.
The Confederate States of America was also a partially recognized independent country during the Civil War.
Some other very small, mostly insignificant "countries" (almost all of which never had any meaningful political power) included the Republic of West Florida, the California Republic, and the Republic of the Rio Grande.
