One devotion of the Catholic Church that is very meaningful to me is the adoration of Christ in the Eucharist. We believe in the real presence of Christ under the disguise of bread and wine. It ain't just crackers and grape juice--Christ is in our midst.
Many parishes have special chapels for 24-hour adoration where 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. Others have special times set aside for adoration.
What a joy to spend an hour with Jesus.
Every Sunday afternoon our parish here in Denver has an Adoration hour for children. My ten year-old daughter (who has felt a calling to be a nun since she was 4) heard God's voice. Afterward she asked the other kids around her if they heard anything and none of them did.
My favorite place for Adoration is Holy Ghost Catholic Church in downtown Denver. Built in 1924 it is a beautiful place to spend time with Jesus.
Here is an example of Holy Hour guidelines from the Adoration Chapel at Christ the King, a Charismatic Catholic parish in Ann Arbor.
More info about Eucharistic Adoration:
Apostolate for Eucharistic Life (Ann Arbor, MI)
Real Presence Eucharistic Education & Adoration Ass'n
Apostolate for Perpetual Eucharistic Adoration
The History of Eucharistic Adoration by Fr. John A. Hardon (pdf)
The Book of Perpetual Adoration by Henri Marie Boudon
Legends of the Blessed Sacrament by Emily Mary Shapcote
The Priest of the Eucharist (St. Peter Eymard)
Holy Orders dedicated to Adoration
Benedictine Sisters of Perpetual Adoration
Holy Spirit Adoration Sisters (The Pink Nuns)
Maronite Monks of Adoration
Poor Clares of Perpetual Adoration
Sisters of Perpetual Adoration