Donatists (named for Donatus Magnus) believed that the authority of a person was correlated to the works of that person. In this particular case, a wave of persecution in 303 through 305 had led some Christians to deny the faith.
The Donatists were looking to see these people permanently removed from the church, and denied positions where they would be performing the sacraments.
Their theology held that the sacraments were not "effective" when performed by "illegitimate" people.
Although Donatism captured a large population of believers (particularly in Africa, see pg 177-178 in "The Rebirth of Orthodoxy"), it was (rightly) rejected by orthodox believers.
It is important to remember that it is God who is "effective" for everything in our lives. There is nothing special about the leader of your church, you, me, or any other person besides Jesus Christ. It is the Father who chooses us and draws us to Him. It is the Holy Spirit who baptizes us into the body of Christ (see all of 1 Corinthians 12). It is Christ who is the payment for our sin debt.