What does the "its" in "its power" refer to in verse 5?
The referent of the genitive-feminine-singular αὐτῆς ("of it") is the genitive-feminine-singular εὐσεβείας ("of godliness")
What kind of power is Paul referring to?
Dave has already provided a discussion of the word translated "power"; I'll offer some thoughts on what power?
The Greek εὐσέβεια is the same word Peter uses to describe how he made a lame man walk, in Acts 3:12. Peter is quite explicit that it was not his own power and ability, but that it was through the power of the Son of God that this miracle was performed, and that:
through faith in his [Jesus'] name hath made this man [the previously lame man] strong (Acts 3:16)
The ability of God (or His invested representatives) to speak and it is so is manifested at the very beginning of the Biblical text:
And God said, Let there be light: and there was light (Genesis 1:3)
This is the power of God, the power by which the lame man was healed, the power that Paul warns many will deny.
How is this power being denied?
Though denial of God's power is not unique to a single era, Paul warned about its prevalence in the last days. Denial of God's power is plainly evident in the "God is dead" movement, as well as in the somewhat less ostentatious "God is silent" movement--it is the idea that either a) there is no God or b) if there is, He cannot now do the kinds of things described in the Bible.
As a 19th-century minister (his name has been lost to history, but his words have not) declared:
there was no such things as visions or revelations in these days; that all such things had ceased with the apostles, (source).
This prejudice against the present was present (no pun intended) in Jesus' day, and He emphatically denounced it:
Let them alone: they be blind leaders of the blind. And if the blind lead the blind, both shall fall into the ditch (Matthew 15:14).
The religious leaders of the day were content to accept what God had revealed in the past, but had little interest in anything God was revealing in the present, especially if it came through the politically-incorrect John the Baptist (Matt 14:4), the uncomfortably-bold Jesus of Nazareth (Matt 15:12), the economically-diminutive Peter (Acts 3:6), or the prison-frequenting Paul.
Had they been more open to their scriptures they may have realized God was less likely to give them new information through the rich and powerful and more likely to employ the humble. They were expecting an Eli and they got a Samuel. Forasmuch as they mocked Jesus' modest origins, they seemed to forget that His ancestor David also came from humble beginnings (see 1 Sam 16:7-11).
Why is it wrong to deny this power?
CS Lewis masterfully illustrated the difference between those who want a comfortable god, and those who want the real God, when discussing Aslan in The Chronicles of Narnia:
"Then he isn't safe?" said Lucy
"Safe?" said Mr. Beaver...."Who said anything about safe? 'Course he
isn't safe. But he's good." (The Lion the Witch and the Wardrobe pp.
Some wish for a tame God whose words can be twisted to mean anything they like, a God who will not contradict them, or ask them to do anything they don't want to do. This is not the God of the bible:
he is like a refiner’s fire, and like fullers’ soap (Malachi 3:2)
He wants to make us clean & pure, but the finished product we can become will never happen if we reject His power in our lives.