As Mormons define it: the authority to act in the name of God.
Now that I look at just that definition, I cringe. Why would God deny his children the ability to act in his name just because they didn't have the Melchizedek or Aaronic priesthood--e.g., be a worthy male Mormon twelve or older.
A thorough look at Hebrews, especially chapter 7, tell us some simple things that show how wrong that is. 7:12 tells us that there is a change in the priesthood and the law. Well, when did the law change? When Christ changed it through his atonement. And so the priesthood changed as well. The same chapter, as well as chapter 5, also tell us that Christ is the high priest and the only one we'll ever need. One of the words used in the original Hebrew to describe Christ's priesthood actually is literally translated as "untransferable inviolate." Which means that only Christ holds the priesthood and it cannot be violated or changed. The reason that Christ's priesthood is described as being after the order of Melchizedek is because Christ, like Melchizedek, lives forever, and is not of the tribe of Aaron. Melchizedek, like so many other people and things of the Old Testament, was simply a symbol of Christ.
So what is the priesthood now? is what Mormons will ask me. Its beautiful, is what it is. The first chapter of Revelation tells us: "To him who loves us and has released us from our sins by his own blood--he has made us a kingdom of priests for his God and Father--to him be glory and power forever."
1 Peter 2:9-10 says "You are the chosen race, the royal priesthood, the consecrated nation, his own people, so that you may declare the virtues of him who has called you out of darkness into his wonderful light; you who were once no people, but are now God's people"
And Romans 12:4-6 and 1 Corinthians 12:27 tell us that we are the body of Christ.
No wonder we're a royal priesthood, a chosen race, a kingdom of priests! We are the body of Christ, because we have his Spirit residing within us! We have Christ's own priesthood and authority! As long as we are in line with his will, his Word, and his love, our faith is our authority! Our faith is our belief! This is why we do not need prophets of the sort of the Mormon church--we have both the prophets of the Bible and we have a direct link to God himself! And apostles? We have their testament in a beautiful book along with the prophets. Is it a coincidence that the Old Testament is the prophets, and the New Testament is the apostles? And every one of us can take time to learn God's word well enough to teach it and evangelize to others. We have all the requirements for being the true church, for having true authority, just by being Christians!
When I was Mormon, it was not that simple. I never saw a miracle done by the priesthood that couldn't either be explained by faith, natural causes, or, worst of all, by Satan's own deceptions. Perhaps that sounds cynical. But I have experienced blessings that only God's goodness in creating this world and setting it in motion in the way he wanted could have blessed me with, and there is no possible way that Satan did that, and I can't believe that this life is all for nothing, just a series of meaningless things that have come together to create this life.
I struggled with the priesthood for the while because I couldn't see how God could be so sexist. When it was in the Old Testament, only the men of the tribe of Levi could hold the priesthood. It was so specific that it was harder to call it sexist. It was for a specific purpose (sacrifice for sins, which was fulfilled in Christ, the ultimate and final sacrifice) and those people, because of their faithfulness at a crucial time, earned the right to the priesthood.
But the Mormon priesthood isn't like that. Its made in such a way that a woman can't even lay her hands on her child's head and heal him or her in the name of Christ, by the priesthood--but a man can. Why was I to be denied the rights that I had as a believer just because I was a woman? I finally got some "answer" because certain women in the temple are allowed to do certain priesthood ordinances for other women because gender separation in those things keeps the men from administering them. And they said that it was so sacred for a woman to hold the priesthood for those ordinances, and I liked hearing that. A good example of the pride that exists in the church, because of its exclusivity.
The answer wasn't actually satisfying, but it felt like God was telling me "don't worry, I know you're struggling with this."
How beautiful was the truth when I found it. How beautiful is the true answer. I cannot help but marvel at what I continue to learn, and how much Christ is easing the bitterness and pride left from my experiences and showing me the beauty of what he offers those who accept the one and only God of love and truth as their own.