Eucharistic adoration is a practice in the Roman Catholic and some Anglican Churches, in which the Blessed Sacrament is exposed to and adored by the faithful. When this exposure and adoration is constant (that is, twenty-four hours a day), it is called perpetual adoration. In a parish, this is usually done by volunteer parishioners; in a monastery or convent, it is done by the resident monks or nuns...This is all based around the Catholic belief that the Eucharist is the body and blood, soul and divinity of Christ. So when you sit with the Eucharist, you are sitting with Jesus Himself. (Much more information can be found here.)
The host is displayed in a monstrance, typically placed on an altar. The Blessed Sacrament may not actually be exposed, but left in a ciborium, which is likewise placed on an altar. This exposition usually occurs in the context of a service of Benediction or similar service of devotions to the Blessed Sacrament. In services of perpetual adoration, parishioners volunteer to attend for a certain period of time, typically an hour, around the clock. Because of the difficulty of maintaining twenty-four hour attendance, many parishes no longer provide perpetual adoration. In many parishes, the Blessed Sacrament is reserved in an enclosed tabernacle so that the faithful may pray in its presence without the need for volunteers to be in constant attendance (as must be the case when the Blessed Sacrament is exposed).
I remember being intensely curious about this practice about three years ago. I ordered and read No Wonder They Call It the Real Presence: Lives Changed by Christ In Eucharistic Adoration which just fueled my curiosity. (I highly recommend that book, by the way. You can read an excerpted chapter here.)
About three weeks after I finished reading the book I had an opportunity for Adoration and the result was ... nothing. Oh, it was peaceful and nice but that was all.
Upon subsequent occasions I have had much more of a connection, words popping into the back of my mind, answers to questions about a course of action to take, the sort of thing that leave my husband shaking his head. He is not exactly disbelieving but just ... shakes his head. He's used to my crazy ways. I don't need adoration for that. It happens quite frequently in front of the tabernacle during Mass.
You can simply sit peacefully with Jesus the entire time, read a devotional book, write in a journal, that sort of thing. It depends on the person. No matter what, the experience is one of peace. That is pretty much a universal result. And if you mention that to my husband he won't move his head at all ... except to nod in agreement!