Adoration of the Blessed Sacrament is a form of worship and devotion to Jesus Christ, present in the consecrated host. Christians around the world have worshiped and prayed in this way for at least 750 years.
Catholics believe that during the Mass which we attend each week (for some of us daily), the priest (during the consecration) speaks these words as he holds the communion host, "...He took bread and gave you thanks. He broke the bread, gave it to his disciples, and said: Take this all of you, and eat it: this is my body which will be given up for you". When the priest says "this my body", it is at that instant when, through the miracle of transubstantiation, the bread and wine which we offer as the bloodless sacrifice to our Lord truly become the Body, Blood , Soul and Divinity of Jesus. It is His true Presence in the form of bread and wine. It is Christ.
Adoration
is when the priest takes a consecrated host, such as the one described above, and places it in a monstrance. (monstrance comes from the Latin "monstrare" to show, to expose to view.) The monstrance is then placed in front of the tabernacle (an ornate box which holds the monstrance and any consecrated hosts) or on the altar of the church or chapel for adoration.
What do you actually do during adoration?
You may sign up to be an "adorer" which allows you to schedule yourself for one or more hours per week to pray before the very presence of Our Lord, exposed in the monstrance. It means that you can have some time alone with Jesus to recite your favorite prayers, read the bible, contemplate acts of faith, hope, charity, thanksgiving, reparation, pray a rosary or do whatever type of prayerful devotion that suits you before Our Lord. You can just sit and say nothing simply keeping Him company, just as you would with a dear friend. There is a custom in many places to hold perpetual adoration where at least one person (often two) spend time before our Eucharistic Lord 24/7.