What is the Catholic concept of grace?

Grace is a participation in the life of God. It introduces us into the intimacy of Trinitarian life: by Baptism the Christian participates in the grace of Christ, the Head of his Body. As an "adopted son" he can henceforth call God "Father," in union with the only Son. He receives the life of the Spirit who breathes charity into him and who forms the Church. - CCC 1997

From the "PENNY CATECHISM, 139" God's grace is a supernatural gift of God, freely bestowed upon us for our sanctification and salvation. The other numbers 140, 141, and 249 are added to aid in the understanding of how grace is obtained and what it does in the sanctification and salvation of a person.

139 What is Grace?
Grace is a supernatural gift of God, freely bestowed upon us for our sanctification and salvation.

140 How must we obtain God's grace?
We must obtain God's grace chiefly by prayer and the holy Sacraments.

141 What is prayer?
Prayer is the raising up of the mind and heart to God.

249 What is a Sacrament?
A Sacrament is an outward sign of inward grace, ordained by Jesus Christ, by which grace is given to our souls.

Definition

GRACE In biblical language the condescension or benevolence (Greek charis) shown by God toward the human race; it is also the unmerited gift proceeding from this benevolent disposition. Grace, therefore, is a totally gratuitous gift on which man has absolutely no claim. Where on occasion the Scriptures speak of grace as pleasing charm or thanks for favors received, this is a derived and not primary use of the term.

As the Church has come to explain the meaning of grace, it refers to something more than the gifts of nature, such as creation or the blessings of bodily health. Grace is the supernatural gift that God, of his free benevolence, bestows on rational creatures for their eternal salvation. The gifts of grace are essentially supernatural. They surpass the being, powers, and claims of created nature, namely sanctifying grace, the infused virtues, the gifts of the Holy Spirit, and actual grace. They are the indispensable means necessary to reach the beatific vision. In a secondary sense, grace also includes such blessings as the miraculous gifts of prophecy or healing, or the preternatural gifts of freedom from concupiscence.

The essence of grace, properly so called, is its gratuity, since no creature has a right to the beatific vision, and its finality or purpose is to lead one to eternal life. (Etym. Latin gratia, favor; a gift freely given.)

Source: Dictionary: GRACE | Catholic Culture

Endnote

From this definition, we then understand the Immaculate Conception of the Virgin Mary and her sinlessness, Arch-Angel Gabriel greeted her, "Hail Full of Grace", as if by her proper name. She was fully sanctified and fully saved.

There are many types of grace. Some God freely gives (e.g., the grace to convert a sinner toward prayer and repentance). Others are merited. There are sanctifying, gratuitous, cooperating, and operating graces (cf. St. Thomas Aquinas's Treatise on Grace in his Summa Theologica).

Read Fr. Réginald Garrigou-Lagrange, O.P.'s introduction to his commentary on the treatise on grace of St. Thomas Aquinas's Summa Theologica. It presents all the various meanings of the word grace (χάρις or "charis" in Greek) as well as the misconceptions (errors) people have historically had regarding grace.

Here is an instructive diagram from the aforementioned book:

Chapter 49 of Fr. Garrigou-Lagrange's Reality treats grace:

…fallen man can, without grace, by God's natural concurrence, know and admit the supernatural truths materially, by an imperfect consent given for a human motive… [F]aith, founded formally on the veracity of God, the author of supernatural life, is impossible without grace.

and

Grace is necessary for knowing supernatural truth, for doing good, for avoiding sin, for disposing man unto justification, for performing each meritorious act, for persevering unto the end.

"Article Two: The Essence Of Grace" shows how the conception of grace of the Nominalists,

who admit in grace only a moral right to eternal life, a right which may be compared to paper money, which, though it is only paper, gives us a right to this or that sum of silver or gold … prepared the way for that of Luther, which makes grace a mere extrinsic imputation to us of Christ's merits.

Thus, Luther's theory of justification is against Catholic doctrine because he denies that sanctifying grace transforms the soul from within. Luther compared grace to snow covering a dunghill, whereas Catholics say of grace, with King David: "Thou shalt sprinkle me with hyssop, and I shall be cleansed: thou shalt wash me, and I shall be made whiter then snow." (Psalm 51:9).

For most Catholic questions there are two places to start:

• Summa Theologica, by Thomas Aquinas
• Catechism of the Catholic Church

So for this question I will use this from the CCC (#2) http://www.vatican.va/archive/ccc_css/archive/catechism/p3s1c3a2.htm

And from Summa Theologica this: http://www.newadvent.org/summa/2111.htm

So there are two types of graces that come from God through Christ and the Holy Spirit, to help heal our sin, to sanctify us:

1. Sanctifying grace is a habitual gift, a stable and supernatural disposition that perfects the soul itself to enable it to live with God, to act by his love.
2. Habitual grace, the permanent disposition to live and act in keeping with God's call, is distinguished from actual graces which refer to God's interventions, whether at the beginning of conversion or in the course of the work of sanctification.

There are two other types of graces that are given by the Holy Spirit to help us to do God's will, which are a subdividing of the habitual grace from above:

1. There are sacramental graces, gifts proper to the different sacraments.
2. There are furthermore special graces, also called charisms after the Greek term used by St. Paul and meaning "favor," "gratuitous gift," "benefit." Whatever their character - sometimes it is extraordinary, such as the gift of miracles or of tongues - charisms are oriented toward sanctifying grace and are intended for the common good of the Church.

There are two types of grace, the first is a sanctifying grace which helps to unite us with God (#1 in the first group) and the second is a gratuitous grace, which allows someone to lead others to God (the second group). This is gratuitous because, much as when Moses was told to lead, he suggested his brother (a better speaker) instead, but, because of God Moses was able to lead, to do more than he expected to do.

So, we look at the cross as an example, where there is a grace that goes between us and God (the vertical part) and the horizontal bar is where we are tied to each other, as we help each others to be with God, so the gift of God has two distinct part, and the RCC differentiates between them.

This leads to a discussion on the RCC's view on justification, but that is beyond the scope of this question, so I will end it here.