The power being spoken of is the visible, audible and physical evidence of the Spirit of Yah dwelling in the body of the believer. The power gives the believer the ability to accomplish supernatural things uncommon to mortal man.
It could show up in Stephen's discourse in Acts Chapters 6:8-10-"And Stephen, full of faith and power, did great wonders and miracles among the people. 6:9 Then there arose certain of the synagogue, which is called the synagogue of the Libertines, and Cyrenians, and Alexandrians, and of them of Cilicia and of Asia, disputing with Stephen. 6:10 And they were not able to resist the wisdom and the spirit by which he spake."
It's central in the operation of the kingdom of Yahweh-1st Corinthians 4:20 says "For the kingdom of God is not in word, but in power."
It also doesn't seem to be apart from the Holy Spirit in the NT context. Acts 1:8 puts it plainly: "But ye shall receive power, after that the Holy Ghost is come upon you: and ye shall be witnesses unto me both in Jerusalem, and in all Judaea, and in Samaria, and unto the uttermost part of the earth."
Quite frankly, I believe it's the kingdom being spoken of in Matthew 24:14 that must, is and will continue to be preached which would usher in "the end"... think about it. The indwelling of the Spirit and associated power saved 3,000 people in one day as recorded in Acts- 2:41. Satan does not want believers to understand this power; evidenced by Paul having to escape Thessaloniki during the cover of night for fear of the synagogue of the Jews-Acts 17:10.
I know it is a kingdom already present on earth which is evidenced by the Ruach ha Kodesh indwelling some believers.