The verses supporting this position start from Acts 1:6-8. The NIV translation:
So when [the apostles] met together, they asked him, "Lord, are you at this time going to restore the kingdom to Israel?"
He said to them: "It is not for you to know the times or dates the Father has set by his own authority. But you will receive power when the Holy Spirit comes on you; and you will be my witnesses in Jerusalem, and in all Judea and Samaria, and to the ends of the earth."
Now, the apostles were talking about a physical and political restoration of Israel as a nation, which was under the authority of the Roman Empire at the time. But Jesus deflected the question and instead told them of the coming Pentecost (See Acts 2), when they would be blessed with the power to share the story of Christ with other nations.
The two aren't the same. One might argue that Jesus' comment was to replace the apostles' expectation of a physical Second Coming with a spiritual one. Which might work out, if most people's idea of the Second Coming weren't based on the book of Revelation, not Acts. From Revelation 19:
I saw heaven standing open and there before me was a white horse, whose rider is called Faithful and True. With justice he judges and makes war. His eyes are like blazing fire, and on his head are many crowns. He has a name written on him that no one knows but he himself. He is dressed in a robe dipped in blood, and his name is the Word of God.
The armies of heaven were following him, riding on white horses and dressed in fine linen, white and clean. Out of his mouth comes a sharp sword with which to strike down the nations. He will rule them with an iron scepter. He treads the winepress of the fury of the wrath of God Almighty. On his robe and on his thigh he has this name written: KING OF KINGS AND LORD OF LORDS.
That's a lot of symbolic language (especially the bit about the sword coming out of his mouth), but the context is that of the physical second coming of Christ. The next several verses expound on his battle with Satan and his forces on earth. Following the battle, the souls of the martyrs "came to life and reigned with Christ a thousand years.... This is the first resurrection." (Revelation 20:4-5) After that "the new Jerusalem" comes down out of heaven, and God's kingdom literally comes to earth for the rest of eternity.
Revelation was written by the apostle John, well after the events recorded in Acts 1. When people talk about the Second Coming, that's what they're usually referring to.