Sometimes you even hear people confusing graceful and gracious, saying that God is graceful. It's understandable because the word looks as if it means, "full of grace" - like when we read in John 1: 14 that Jesus was "full of grace and truth."
The dictionary, however, clearly explains the difference between the two: graceful refers to, "elegance, poise, physical beauty and fluidity of movement" (like a ballerina), while gracious refers to, "kindness, warmth, courtesy and generosity of character." Graceful describes how someone moves or looks, while gracious describes how someone treats others. The dictionary links gracious not only with kindness and generosity, but also with forgiveness, mercy and favour.
So it's understandable that the Hebrew words hanûn, hanan and hēn are often translated as mercy or compassion in the Old Testament; they express that God shows us kindness and favour, even when we don't deserve it.
But in a broader sense, Yahweh Hannûn (or Channun) expresses that God is gracious towards us - not only that He forgives us when we do wrong, but also that He's kind and generous towards us, even when we've done nothing in particular to deserve His favour. Grace in this sense is truly "undeserved kindness."
The priestly blessing in Num 6: 25 says, "May the Lord be gracious to you," and throughout the Old Testament we see different people praying for and depending on the grace and favour of God. The same understanding continues into the New Testament, where the Greek word charis is similarly used to mean both favour and grace, or even free gift.
The Jerusalem Council (Acts15: 11) stated clearly that we are saved by the grace of God, and not through our own efforts in keeping the Law. Paul's letters to New Testament believers continually unpack this fundamental truth, summing it up well in Eph 2: 8 - it is because of God's grace that you are saved through faith and not through anything that you yourselves have actually done; salvation is a free gift.
But the grace of God goes even beyond saving us from sin and death. It's by grace that we can approach the throne of God and come right into His presence (Heb 4: 16.) It's also God's grace that helps us to persevere in times of difficulty (in 1 Cor 12: 9. for example.)
And even though God's grace is sufficient to save us when we first encounter Him, we discover the astonishing truth that we can continually grow in God's grace - in experiencing it ourselves and in extending it to others - even as we also continue to grow in our knowledge of God. (2 Peter 3: 18) Yes, it's something we can grow in, and the Lord continually gives us more grace as we also keep on growing in knowing Him.