I would like to write about the Franciscan Order and then briefly describe two other monastic movements, while showing how Franciscans were and are different. For example, some emphasize the eremitical aspect of praying, fasting, meditating, while others emphasize missionary, pastoral, and educational efforts.

An excellent and accessible source for the life of St. Francis, founder of the Franciscans, is Paul Sabatier's The Life of St. Francis of Assisi. It was one of the first books to apply modern, objective techniques of scholarship to a Roman Catholic saint.
The Franciscan order is most notable for its vows of poverty. As Sabatier outlines, Francis did not want to found a typical monastic order in which the brothers would own and live in a monastery. Instead, he envisioned them as a mendicant order, one that would raise all its funds through begging. This would be successful, St. Francis thought, because the needs of his brothers would be extremely simple—bare feet, rough, ragged robes, just enough food to survive, and rough huts for housing. Begging and owning nothing would help reinforce their humility. It would also emphasis the reality that all provision comes from God. This was a literal living out of the line in the Lord's Prayer, "give me this day my daily bread."
Francis envisioned his followers living in joyful poverty and did not intend for them to be focused on their sin (he himself was given to flagellation to rid himself of material desires but later apologized to his body for what he had done to it). He himself found joy in a poor life and wanted his followers to experience and express the beauty and joy of God's creation. He also wanted his followers to serve the poorest of the poor with joy and gratitude. He was notable himself for his charity toward lepers, a feared and despised group in his society.
As Francis was dying, he did agree to allow the Franciscans to become more like a typical monastic order, though he had resisted such a move earlier. While there has always been a tension between the order's vows of poverty and communal ownership of property, the severity of Francis's vision has been relaxed. The monks do live in typical monasteries. They still, however, emphasize simple, humble living, as well as serving the poor and blessing the animals and creation. The current pope, St. Francis, exemplifies many ideals of the Franciscans. For details of some of the difficulties living in a order based on charity in times past, you might look at the book Galileo's Daughter, by Dava Sobel. 
The Carthusians are an eremetical order you might focus on to contrast with the Franciscans. The Carthusians are contemplative brothers who live together in a monastery, focusing on lives of prayer, solitude, and silence. 
The Jesuits, an order founded in the sixteenth century, are known for their global missionary work and their emphasis on education.