We serve a Holy God. Isaiah made this pretty clear when he wrote down the words of the seraphs: "Holy, holy, holy is the LORD Almighty." In the Old and New Testaments, holiness is described in a variety of ways. William Greathouse wrote a really good book on this subject. If I tried to write a book on this subject as a blog post, no one would read it. So don't worry, you're not in for a book if you continue reading!
At least 2 ideas of holiness can be found in the Old Testament. The first is holiness as it was connected to ritual purity. This is strongly connected to separation from all things evil or "unclean." The second idea, usually when it's used referring to God, is the idea of a "divine otherness." A "holy God" means that God is divinely something other than humans and creation. So, if God is "holy, holy, holy" he is something much greater and more divine than humanity can fathom.
In the New Testament, Jesus reinterprets holiness in many ways. In the life of Jesus, the "divine other" becomes humans. He also shows that holiness can't be made "unholy" by coming in contact with the "unclean." Instead, Jesus goes and eats with tax collectors and sinners. By his presence with them, it is the "unclean" who are forgiven and made whole again. The Sermon on the Mount explains in greater detail what holiness looks like lived out. Jesus summarizes holiness in the two greatest commandments: love God and love others.
It seems that throughout the Bible, "holiness" is often connected to "righteousness" and "justice." Connecting to these terms expands the description of holiness to include living in right relationship with God and others. It means facing the justice of God in regard to our sins, and being made righteous despite these sins. It means caring for the blind, orphaned, widowed, lame, oppressed, and all those who can't care for themselves. But it's not works-righteousness. Holiness is derivational. That means that we are holy only to the extent that God makes us holy. The closer we are to the "divine other," the more holy He makes us. Picture again the scene from Isaiah 6.
Holiness becomes confusing to most of us when we start speaking of it in terms of "Christian Perfection." This is mostly because the word "perfection" throws us for a loop. When most of us hear this word we think "without flaw or mistake." If a person confesses to have experience "Christian perfection," it would seem in our minds that this person would no longer make [stupid] mistakes. No more open-mouth-insert-foot sort of situations. The problem is, none of us have ever met someone who is without flaw!
However, the "perfection" that actually goes with "holiness" is not "flawlessness." Instead, it's more like "absolutely embodying the definition or description of an ideal." ((I didn't make this up...it's in the dictionary!)) So, to say that Christ was perfect, we mean that he absolutely embodied the ideal relationship with God. As Christians, when we talk about being perfected by the Holy Spirit, we actually mean that God is shaping us to absolutely reflect an ideal relationship with God. This is what holiness is.
Holiness is having a perfect relationship with God and others. It means embodying the ideal - perfect love and absolute surrender. It doesn't mean that we'll stop making mistakes like tripping over our own feet or accidentally messing up a recipe. It means that we will live in total surrender to our Holy God.