Cessationism is a doctrine that spiritual gifts such as speaking in tongues, prophecy and healing ceased with the Apostolic Age (1st century). The belief among cessationists is that supernatural manifestations like this served to authenticate the message of those who preached the gospel at that time. But these manifestations are no longer needed or present today.
The problem with this doctrine, is there is simply NO scriptural basis for believing it.
When the Holy Spirit was poured out on the Day of Pentecost in Acts 2, Peter saw fit to quote Joel 2:
And afterward, I will pour out My Spirit on all people.
Your sons and daughters will prophesy,
your old men will dream dreams,
your young men will see visions.
Even on My menservants and maidservants,
I will pour out My Spirit in those days
The context of these verses from Joel is at the restoration of all things. Note the words "And afterward, I will pour out My Spirit on all people".
After what? The previous statement says:
Then you will know that I am in Israel,
that I am the Lord your God,
and that there is no other;
never again will my people be shamed.
Peter, in quoting these words, was demonstrating that this manifestation of the Spirit that they were experiencing was a foretaste of what is to come - when His kingdom is established on earth.
This was not the end of something, but the beginning.
The author of Hebrews confirms this by his statement that these manifestations of the Spirit are a foretaste of "the coming age":
It is impossible for those who have once been enlightened, who have tasted the heavenly gift, who have shared in the Holy Spirit, who have tasted the goodness of the word of God and the powers of the coming age — and then have fallen away—to be restored to repentance.