Not seeking to start a fight or discussion. OP asked a question, so here is my answer. Not seeking discussion with "cessasionist" either, just FYI. But I just have to let OP know this, and then it's up to him what he'll do with it.
OP, do with this as you want. Best advice I can give you - with regards to all the different answers given - is to pray and ask the Lord to show you what is the truth and what is not.
Acts 1:7-8 (NIV):
7 He said to them: “It is not for you to know the times or dates the Father has set by his own authority. 8 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.”
What is the meaning of "power" in verse 8?
The power of the Holy Spirit.
Was this "power" only for the apostles in the first-century church, or has it been available for every Christian up until now?
Available to every christian. In God's timing, however. Jesus is the Baptizer. We can't force Him to do it whenever we want it. Doesn't work like that. Also, people who deny the existence of Holy Spirit baptism today are (in my opinion) usually just ignorant because they have not experienced the power of the Holy Spirit in their own lives or churches, or (oftentimes) it may be pride, where they are offended at the mere thought of someone else getting baptized in the HS while they themselves have been converted for decades and never received. But there is no need for that. God is not a respecter of man. You don't get baptized as a reward.
You get endued with power from on high, power to be better equipped to do the work of God. Signs and wonders will follow you, and you will become a miracle worker, by the grace of God, by the power of the Holy Spirit.
Lastly, is there any Scripture-based way to tell if one is in possession of this "power"?
There sure is. Read through acts and you will see. The initial evidence of someone getting baptized in the Holy Spirit is that they will (initially) begin to speak in a new tongue. This is not something that can be learned or imitated; if someone does that, it's not the real thing. The real thing is truly God giving you the utterance. So it's Him giving you the utterance to speak; not you trying to speak it.
Note I said "initial" - as this is the sign that shows HS baptism has taken place; however, there can be quite some time between that initial tongues and later speaking in tongues. Also, there are different kinds of tongues; there is one that is meant as a sign to unbelievers (which is in a human language but unknown to the speaker) and there is the other one which is the Holy Spirit giving you the utterance to speak a heavenly language, not unto people but unto God. This prayer is according to the perfect will of God, even though you probably won't always know what you are saying (though you may receive interpretation).
It is extremely powerful, praise the Lord. I've seen demons shivering and crying during deliverance as the pastor started speaking in tongues. God is so wondrous.
Anyway, I also want to add that there is such a thing as demonic tongues. Satan is a liar and a wannabe copycat of God, he tries to rip off and pervert everything. That's why you shouldn't go out and let just anyone put their hands on you because it could actually be a witch transferring demonic spirits onto you, and they could even give you a demonic (counterfeit) tongue.
How can a Christian obtain this "power"?
Different ways, as God is sovereign and He can do it as He sees fit. But I would suggest asking, seeking, knocking. You know that even our earthly fathers know how to give good gifts. How much more will your Heavenly Father give the Holy Ghost to them that ask Him? You have to be patient and humble, though. Some people received it relatively quickly, some did not even have to ask at all, and other have waited for 5 years.
You have to keep in mind that it is God empowering you for service; and also that it is the Promise of the Father. That means promise to you and me, all of us. So just let Him know that you desire this gift, draw near to Him, and let the Holy Spirit sanctify your heart. And live a holy life! God is holy, and He commands us to be holy as well. God has not changed.