The month of June is traditionally dedicated to The Sacred Heart of Jesus, a private devotion revealed to us by Saint Marguerite Marie, a French nun during the 17th century.

From Fisheaters.com

The heart has always been seen as the "center" or essence a person ("the heart of the matter," "you are my heart," "take it to heart," etc.) and the wellspring of our emotional lives and love ("you break my heart," "my heart sings," etc.) Devotion to the Sacred Heart of Jesus is devotion to Jesus Christ Himself, but in the particular ways of meditating on his interior life and on His threefold love -- His divine love, His burning love that fed His human will, and His sensible love that affects His interior life. Pope Pius XII of blessed memory writes on this topic in his 1956 encyclical, Haurietis Aquas (On Devotion To The Sacred Heart).Below are a few excerpts which help explain the devotion:

 

54. ...the Heart of the Incarnate Word is deservedly and rightly considered the chief sign and symbol of that threefold love with which the divine Redeemer unceasingly loves His eternal Father and all mankind.

55. It is a symbol of that divine love which He shares with the Father and the Holy Spirit but which He, the Word made flesh, alone manifests through a weak and perishable body, since "in Him dwells the fullness of the Godhead bodily."

56. It is, besides, the symbol of that burning love which, infused into His soul, enriches the human will of Christ and enlightens and governs its acts by the most perfect knowledge derived both from the beatific vision and that which is directly infused.

57. And finally -- and this in a more natural and direct way -- it is the symbol also of sensible love, since the body of Jesus Christ, formed by the Holy Spirit, in the womb of the Virgin Mary, possesses full powers of feelings and perception, in fact, more so than any other human body.

58. Since, therefore, Sacred Scripture and the official teaching of the Catholic faith instruct us that all things find their complete harmony and order in the most holy soul of Jesus Christ, and that He has manifestly directed His threefold love for the securing of our redemption, it unquestionably follows that we can contemplate and honor the Heart of the divine Redeemer as a symbolic image of His love and a witness of our redemption and, at the same time, as a sort of mystical ladder by which we mount to the embrace of "God our Savior."

59. Hence His words, actions, commands, miracles, and especially those works which manifest more clearly His love for us -- such as the divine institution of the Eucharist, His most bitter sufferings and death, the loving gift of His holy Mother to us, the founding of the Church for us, and finally, the sending of the Holy Spirit upon the Apostles and upon us -- all these, We say, ought to be looked upon as proofs of His threefold love.

60. Likewise we ought to meditate most lovingly on the beating of His Sacred Heart by which He seemed, as it were, to measure the time of His sojourn on earth until that final moment when, as the Evangelists testify, "crying out with a loud voice 'It is finished.', and bowing His Head, He yielded up the ghost."Then it was that His heart ceased to beat and His sensible love was interrupted until the time when, triumphing over death, He rose from the tomb.

61. But after His glorified body had been re-united to the soul of the divine Redeemer, conqueror of death, His most Sacred Heart never ceased, and never will cease, to beat with calm and imperturbable pulsations. Likewise, it will never cease to symbolize the threefold love with which He is bound to His heavenly Father and the entire human race, of which He has every claim to be the mystical Head.

Devotion to the Sacred Heart has two elements: consecration and reparation:

• We consecrate ourselves to the Sacred Heart by acknowledging Him as Creator and Redeemer and as having full rights over us as King of Kings, by repenting, and by resolving to serve Him.
   

• We make reparations for the indifference and ingratitude with which He is treated and for leaving Him abandoned by humanity.

To carry out these general goals of consecration and reparation, there are quite specific devotions authorized by the Church.