Continuationism takes its name and much of its argument from what it opposes. Cessationism, historically strong in Reformed and other confessional Protestant traditions, held that the most dramatic gifts described in 1 Corinthians 12, prophecy, tongues, healing and miracles among them, were given specifically to authenticate the apostles and the church's founding generation, and that they ceased once that founding work was complete and the New Testament canon finished. On this reading, the gifts were a scaffold removed once the building, Scripture and the established church, stood on its own.
Continuationists answer that the New Testament nowhere states an expiration date for the gifts, that Paul writes about them as ordinary features of congregational life rather than as founding-era exceptions, and that restricting their operation to the first century is an inference read into the text rather than a claim the text itself makes. Pentecostal and Charismatic Christians add their own ongoing experience of healing, prophecy and tongues in worship as living evidence against the cessationist claim that these gifts stopped.
The argument has run for over a century without resolution and divides Protestant Christianity roughly along the same lines that divide Pentecostal and Charismatic churches from confessional Reformed and other cessationist traditions, making it one of the clearest doctrinal fault lines within evangelical Protestantism specifically, distinct from the older divisions over sacraments or church government.