Here's what grace means from an LDS perspective: Grace justifies us, it sanctifies us, and it grants us God's power to become stronger.
(1) Justification - Through repentance and a sincere desire to gain the help of Christ's atonement in our lives to do better, we are justified--declared innocent of the sins we have committed.
(2) Sanctification - A purification process that removes from us the effects of sin and gives us hope and optimism to succeed and a realization that God loves us as we are but sees us as we can become.
(3) Empowerment - God's grace--and Christ's atonement--only remove both the punishment for sin and the effects of sin in our lives, they also empower us to become more able to resist committing further sin, a process by which we become more and more like God.
I think that facets 1 and 2 above are very similar concepts in all Christian denominations. Where the Church of Jesus Christ of Latter-day Saints seems a bit unique is in the idea that grace empowers us to become better--in other words, to become like our Heavenly Parents.
For what it might be worth, this sermon is, I think, an excellent explanation of how sanctification and justification are similar, and how they are different.