I’m sure that there are those who can answer much better than I. Devotion to (fill in the blank) means that you give special attention in your prayer life to that aspect of Christ’s life/death/resurrection or to a particular saint. So, first I’ll mention some devotions to Our Lord. If one is devoted to the Blessed Sacrament, that means they spend extra time meditating upon the Blessed Sacrament, often spending many hours in adoration, etc. Some are devoted to the Stations of the Cross, and if they are they might recite it daily. They might even make a pilgramage to Jerusalem to “walk the steps of Jesus”. Two other popular devotions to Christ are the Divine Mercy and the Sacred Heart. Both are based on private revelations from Christ to nuns. There was a chaplet given to St. Faustina which pleads for God’s Divine Mercy upon the world (see my signature for a small bit). For the Divine Mercy devotion, one would read St. Faustina’s Diary of Divine Mercy (it’s amazing), keep a picture our Lord requested to represent His Mercy, and say the chaplet Our Lord gave to St. Faustina every day. The Sacred Heart devotion concentrates more on the tender love of the Sacred Heart of Jesus. There are prayers that go with this devotion too and a special obligation to attend the Mass on the first Friday of nine consecutive months. In both Divine Mercy and the Sacred Heart devotions, Our Lord made certain promises for those who follow the devotions. I must state again that these are private revelations and the Church does not require anyone to believe in them. However, those devotions (and saints) have been thoroughly investigated by the Church and it has been found that they do not conflict with the teaching of the Church. Furthermore, I would say that several popes have encouraged these devotions by establishing feast day’s in their honor.
Devotion to saints is a bit different. People often find at least one (in my case, many) saints whose life inspires them or that they can relate to. They may take that saint as a patron in hopes that the saint will intercede for them to God. My patron is St. Teresa of Avila, a great saint and Doctor of the Church. She is known as the Doctor of Prayer. Since I wanted to have a rich prayer life and since Teresa had such a personal relationship with Our Lord, I chose her. She was quite a character. Devotion to the Blessed Mother is another important (but not required) aspect of Catholic life. I cannot describe how much closer to Christ I have become since becoming devoted to Our Lady. Devotion to Our Lady might include the daily recitation of the Rosary, wearing a Miraculous Medal, remembering her sorrows. Offering up prayers to Our Lord through Our Lady is most effective. The Bible says that the prayers of the righteous avail much and Mary is very righteous in the Lord’s eyes (as are all saints, but Mary is special, for obvious reasons).
I hope this helps. Sorry for the long post… I am actually in the process of increasing my devotions. I think the most important point to remember is that a devotional life, is meant to always bring us closer to Christ, to increase our love for Him, and our love for our neighbor.
PS. If there is a particular devotion you are interested in, let us know, I’m sure that among all the posters here that you’ll find someone who has done it and can tell you all about it.