This question is a bit tricky, depending on whether one considers failed colonies.
The earliest of the Spanish colonies, San Miguel de Gualdape, was established in 1526 on land that is now either in Georgia or South Carolina. The next year, Spain tried again, near St. Petersburg, Florida. Spain established a colony at St. Augustine in Florida in September of 1585. In the west, Spain tried to colonize in 1598 what is now Santa Fe, NM.
France tried to establish colonies at least three times in Maine, South Carolina, and Florida in the late 16th and early 17th centuries.
The English had a failed colony at Roanoke in 1585 and tried again in 1587. England founded two American colonies in 1607, in Popham, Maine and Jamestown, Virginia. Popham was a failure, but Jamestown survived, barely. In 1620, The Mayflower brought English to Massachusetts, and the subsequent Plymouth and Massachusetts Bay Colonies would grow quite large and prosperous.
It can be said, then, that attempts by European countries to colonize America began in the early sixteenth century. Reasons for colonization are generally for two reasons: to gain economic and geographic power for the European nations and to seek a respite from religious persecution in Europe.