Catholics believe that the Eucharist is the real presence of Jesus Christ. Eucharistic Adoration is the worship, praise, and adoration of Jesus Christ present in the Holy Eucharist. The Eucharist is displayed in a special holder called a monstrance, and people come to pray and worship Jesus continually throughout the day and often the night. Christ loves us without limit, and offers Himself to us in the Holy sacrament of the Eucharist. Eucharistic adoration is a way for us to give Jesus love and worship in return.
For more information on Eucharistic Adoration and on the real presence of Christ in the Eucharist, including articles and resources, go to the website of the Real Presence Eucharistic Education and Adoration Association.
From the Directory on Popular Piety and the Liturgy (2001)
Congregation for Divine Worship and the Discipline of the Sacraments:
164. Adoration of the Blessed Sacrament is a form of Eucharistic cult which is particularly widespread in the Church and earnestly recommended to her Pastors and faithful. Its initial form derives from Holy Thursday and the altar of repose, following the celebration of the Coena Domini Mass. This adoration is a most apt way of expressing the connection between the celebration of the memorial of the Lord’s Sacrifice and his continued presence in the Sacrament of the Altar. The reservation of the Sacred Species, so as to be able to administer Viaticum to the sick at any time, encouraged the practice among the faithful of recollection before the tabernacle and to worship Christ present in the Sacrament (175).
Indeed, this worship of adoration has a sound and firm foundation,  especially since faith in the Lord’s real presence has as its natural consequence the outward and public manifestation of that belief. Therefore, the devotion prompting the faithful to visit the blessed sacrament draws them into an ever deeper share in the paschal mystery and leads them to respond gratefully to the gift of him who through his humanity constantly pours divine life into the members of his Body.  Abiding with Christ the Lord, they enjoy his intimate friendship and pour out their hearts before him for themselves and for those dear to them and they pray for the peace and salvation of the world. Offering their entire lives with Christ to the Father in the Holy Spirit, they derive from this sublime colloquy an increase of faith, hope, and charity. Thus they foster those right dispositions that enable them with due devotion to celebrate the memorial of the Lord and receive frequently the bread given us by the Father. (176)
165. In adoration of the Blessed Sacrament, which can take different forms, several elements deriving from the Liturgy and from popular piety come together and it is not always easy to determine their limits (177):
The faithful should be encouraged to read the Scriptures during these periods of adoration, since they afford an unrivalled source of prayer. Suitable hymns and canticles based on those of the Liturgy of the Hours and the liturgical seasons could also be encouraged, as well as periods of silent prayer and reflection. Gradually, the faithful should be encouraged not to do other devotional exercises during exposition of the Blessed Sacrament (180). Given the close relationship between Christ and Our Lady, the rosary can always be of assistance in giving prayer a Christological orientation, since it contains meditation of the Incarnation and the Redemption (181).
Stay with me, Lord
Prayer of St. Pio of Pietrelcina after Holy Communion
Stay with me, Lord, for it is necessary to have You present so that I do not forget You.
You know how easily I abandon You.
Stay with me, Lord, because I am weak and I need Your strength, that I may not fall so often.
Stay with me, Lord, for You are my life, and without You, I am without fervor.
Stay with me, Lord, for You are my light, and without You, I am in darkness.
Stay with me, Lord, to show me Your will.
Stay with me, Lord, so that I hear Your voice and follow You.
Stay with me, Lord, for I desire to love You very much, and always be in Your company.
Stay with me, Lord, if You wish me to be faithful to You.
Stay with me, Lord, for as poor as my soul is, I want it to be a place of consolation for You, a nest of love.
Stay with me, Jesus, for it is getting late and the day is coming to a close, and life passes; death, judgment, eternity approaches. It is necessary to renew my strength, so that I will not stop along the way and for that, I need You. It is getting late and death approaches, I fear the darkness, the temptations, the dryness, the cross, the sorrows. O how I need You, my Jesus, in this night of exile!
Stay with me tonight, Jesus, in life with all its dangers. I need You.
Let me recognize You as Your disciples did at the breaking of the bread, so that the Eucharistic Communion be the Light which disperses the darkness, the force which sustains me, the unique joy of my heart.
Stay with me, Lord, because at the hour of my death, I want to remain united to You, if not by communion, at least by grace and love.
Stay with me, Jesus, I do not ask for divine consolation, because I do not merit it, but the gift of Your Presence, oh yes, I ask this of You!
Stay with me, Lord, for it is You alone I look for, Your Love, Your Grace, Your Will, Your Heart, Your Spirit because I love You and ask no other reward but to love You more and more.
With a firm love, I will love You with all my heart while on earth and continue to love You perfectly during all eternity.