What is the power that Christians possess? There are some verses that use the terminology &quot;power&quot;, and I heard it with different preachers that the Christians possess power.
Here are some examples:

and my speech and my message were not in plausible words of wisdom, but in demonstration of the Spirit and of power, so that your faith might not rest in the wisdom of men but in the power of God. (1 Corinthians 2:4-5, ESV)
And Stephen, full of grace and power, was doing great wonders and signs among the people. (Acts 6:8, ESV)

What is the power that Christians possess?
The following comes from a mainly Calvinist view point:

The power you are looking for is the power granted to Christians by the Holy Spirit. Probably one of the most clear examples of this was in Acts 2 on the day of Pentecost. 


  5 Now there were staying in Jerusalem God-fearing Jews from every nation under heaven. 6 When they heard this sound, a crowd came together in bewilderment, because each one heard their own language being spoken. 7 Utterly amazed, they asked: “Aren’t all these who are speaking Galileans? 8 Then how is it that each of us hears them in our native language? 9 Parthians, Medes and Elamites; residents of Mesopotamia, Judea and Cappadocia, Pontus and Asia,[b] 10 Phrygia and Pamphylia, Egypt and the parts of Libya near Cyrene; visitors from Rome 11 (both Jews and converts to Judaism); Cretans and Arabs—we hear them declaring the wonders of God in our own tongues!” 12 Amazed and perplexed, they asked one another, “What does this mean?”  (NIV)


This was the first manifestation of the power of the Holy Spirit on these new Christians. Much of the book of Acts continues to chronicle this power that they had been granted. 

There is now some debate how this power manifests in the modern 21st century. However, it is no doubt still active. Even those (like myself) that are cessationists believe that the spirit is still active in our daily lives, but that he manifests his power differently than he did in the early church.

For example many believe that one of the greatest examples of the power of the spirit is his working on the hearts and minds of God's people to perform both regeneration and sanctification.