If you'd like, you can check out the
Philokalia, which is essentially an anthology of Orthodox monastic texts compiled over the centuries.
Sayings of the Desert Fathers and St. John Climacus's
Ladder of Divine Ascent are also excellent.
To give a brief overview of the
Orthodox monastic tradition, many monks live together in either a large communal monastery (such as the Valaam Monastery, or St. Catherine's at Mt. Sinai, as well as several monasteries on Mt. Athos), in smaller groups known assketes, and some even live on their own in isolation as hermits. Regardless of the living arrangements, all monastics leave "the world", as we call it, and enter into the monastic life to devote their entire lives to deepening their relationship with Christ. They take literally the command to sell all they have, give the money to the poor, take up their cross, and follow Him. They do this not because they think the world is evil--even if it is a great source of temptation, and it's easy to be like the seed that gets choked out by the thorns of worldly cares. Rather, they do it because they love Christ, and by their prayers and ascetic way of life, they hope to pray for the entire world, doing whatever they can to sanctify themselves and everyone around them to God.
A great example of the life of a monk is St. Antony the Great--one of the first great monks, and one of the biggest players the Church (and Christianity as a whole) has ever known in the realm of monasticism. He essentially founded monasticism in Egypt, or at least did a LOT of work to spread it. You can read his life
here. (CTRL+F to "
Athanasius: LIFE OF ANTONY" to skip right to where the account of his life starts off.)
As Dunemeister already pointed out, we Orthodox have a spiritual practice known as
hesychasm. It's also said that, in Orthodoxy, everyone has a little bit of monasticism in them, regardless of whether we're married, single, celibate, young, old, clergy or laity.