Outside the imaginative confines of the Church (and nearly as often inside) the concept of Grace is one of the more misunderstood Christian ideas. The word is thrown around a lot, without explanation. Outside of the Church, Grace usually means something like "elegance" or "poise." It is a quality that a person can have, which can either be learned and practiced, or is inborn: a gift of genetics. This is something like the Grace we mean, I think, in that the Grace of God is a free gift, and yet we can lean into it, practice it, get better at both accepting it and bestowing it on those around us. But as for what it actually is, it is more than mere elegance.
From a certain angle, Grace is like mercy -- it is undeserved. We don't earn God's grace. But that's just a piece, because I don't think it is primarily about forgiveness from deserved punishment. When you give a gift to a friend, not out of any sense of obligation, but because the giving will make both of you happy, that is grace. Grace is my daughter (NOT coincidentally named Grace) when she dives headfirst into a snowdrift, wild with a mad joy at just being alive. Divine Grace, as Father Richard Rohr points out below, keeps the planets spinning in their orbits for no good reason "except love alone." The crystallization of this idea, though it can come through many traditions and sources, is in my mind the finest contribution Christian thought has made to the world imagination. Here's Father Rohr, who puts it better than I do.
-Ben (sorry for the long post!)
God Is Eternally Giving Away God
I think grace, arising from God's limitless love, is the central theme of the entire Bible. It is the divine Unmerited Generosity that is everywhere available, totally given, usually undetected as such, and often even undesired. This grace was defined even in the old Baltimore Catechism as "that which confers on our souls a new life, that is, a sharing in the life of God himself [sic]."  We always knew it on paper, but much less in experience and conviction.
In the parable of the watchful servants (Luke 12:35-40), God is actually presented as waiting on us--in the middle of the night! In fact, we see God as both our personal servant inside our house and the divine burglar who has to "break through the walls of [our] house." That's really quite extraordinary and not our usual image of God. It shows how much God--the "Hound of Heaven," as Francis Thompson says--wants to get to us and how unrelenting is the work of grace.
Unless and until you understand the biblical concept of God's unmerited favor, God's unaccountable love, most of the biblical text cannot be interpreted or tied together in any positive way. It is, without doubt, the key and the code to everything transformative in the Bible. People who have not experienced the radical character of grace will always misinterpret the meanings and major direction of the Bible. The Bible will become a burden, obligation, and weapon more than a gift.
Grace cannot be understood by any ledger of merits and demerits. It cannot be held to patterns of buying, losing, earning, achieving, or manipulating, which is where, unfortunately, most of us live our lives. Grace is, quite literally, "for the taking." It is God eternally giving away God--for nothing--except the giving itself. I believe grace is the life energy that makes flowers bloom, animals lovingly raise their young, babies smile, and the planets remain in their orbits--for no good reason whatsoever--except love alone.