During the Vietnam War, roughly 2/3rd's of Americans enlisted in the military were voluntarily. The other 1/3rd was done through conscription (aka a military draft). These men were selected through the Selective Service System. Men aged 18-25 were put into a lottery type system in which men were randomly selected. From there, they would report to the local authorities who would help them get their affairs in order.
One of the biggest reasons why this draft upset Americans was because a majority of the individuals who were required to serve were from poor or lower class families. Individuals from wealthy backgrounds or who knew politicians were able to get exemptions from their service.