The draft for the 1960 season was actually conducted in late 1959, shortly after the formation of the league.
Thereafter, American Football League drafts were conducted separately from the rival NFL through 1966.
Starting in 1967, after the NFL agreed to merge with the AFL, the two leagues conducted a "common draft", which was in turn replaced with the modern NFL Draft in 1970, upon the completion of the AFL/NFL merger.
The first and one of the most prominent of these was LSU All-American Billy Cannon, who went on to become an AFL All-Star both as a running back with the Houston Oilers and as a tight end with the Oakland Raiders.
Other greats signed by the AFL in the years before the common draft included Abner Haynes and Johnny Robinson (Dallas Texans); Jim Otto (Oakland Raiders); Lance Alworth, John Hadl, and Ron Mix (San Diego Chargers), Lionel Taylor (Denver Broncos); Billy Shaw (Buffalo Bills); Larry Grantham (New York Titans); Matt Snell and Joe Namath (New York Jets); Nick Buoniconti (Boston Patriots) and many others.