There were a lot of different orders of monks in the Middle Ages (still are, in fact), but the primary monasteries in England consisted of:
Dominicans: Dominicans are about preaching and doctrinal conformity. They were (no surprise) the order behind the inquisition, with the intent to rule out any doctrine that didn’t abide strictly by received Catholic theology. “Domingo de Guzman (around 1170-1221), a Spanish priest travelling with his bishop Diego of Osma, encountered by chance Cistercian monks who tried to bring the Cathars of Southern France back to the Catholic Church. He saw the deficiencies of their attempts and decided to do a better job, by walking and dressing humbly, listening to and talking with people, being aware of contemporary developments, and first of all preaching the Gospel. He gathered a band of priests around him. After the Fourth Lateran Council of 1215, at which preaching and confession had been stressed as important elements of priestly activities, Dominic got papal approval in 1216 from Honorius III for founding a new order to meet these needs, the “Ordo Praedicatorum”, using the Rule of St. Augustine. In 1207 Dominic had already founded a nunnery at Prouille, near Toulouse. He sent the members of his small group to all universities to study theology.” http://home.kpn.nl/otto.vervaart/dominican_order.htm
In addition ,from a reader’s comment: “Dominicans are friars, who are not cloistered, but instead tasked with wandering about preaching. They were mendicants, like the Franciscans and Carmelites and others. Friars are not the same as monks. It is an easy mistake to make, one I was guilty of as well until a fellow re-enactor needled me about it.
The Dominicans founded their houses or priories near towns, where there was demand for their services. They were very much part of the community and were very important in the 13th century revival of Catholicism. (well maybe not quite revival, but they spread the gospel very well)”
Benedictines: The first Benedictine house was established in 529 AD in Italy by St. Benedict. While many similar houses were established in the next 1000 years, there is no ‘mother house’ in this order, with each community of monks being independent and autonomous. “The Rule of St Benedict requires candidates for reception into a Benedictine community to promise solemnly stability (to remain in the same monastery), conversatio morum (an idiomatic Latin phrase suggesting “conversion of manners”), and obedience (to the superior, because the superior holds the place of Christ in their community). This solemn commitment tends to be referred to as the ‘Benedictine vow'”. http://en.wikipedia.org/wiki/Order_of_Saint_Benedict They are known as the ‘black monks’ for their dark robes.
Augustinians: They’ve got an official web site! http://www.osanet.org/home.asp?language=EN
From the web site: “The concrete historical situation of the Christian world in the thirteenth century caused the Apostolic See to promote the formation of the Mendicant Orders. In this context Pope Innocent IV, with the Rule of St. Augustine as his basis, determined the norms according to which some groups of hermits of Tuscany were united and organized. From here the Order of Hermits of Saint Augustine arose juridically in March 1244. This first nucleus was consolidated and amplified by the aggregation of other religious groups in the Grand Union of 9 April 1256 promoted by Pope Alexander IV. This singular intervention of the Apostolic See determined its activity in a specific way in service to the Universal Church. [Constitutions of the Order 2008, n. 3]” The focus on the Augustinian order is learning and knowledge, thus their collection of libraries and education and missionary work. http://en.wikipedia.org/wiki/Order_of_Saint_Augustine
Cistercians: An offshoot of the Benedictines, “the Cistercian order came about through the actions of a group of break-away monks who left their monastery of Molesme in Burgundy, in order to try to adhere more strictly to the Rule of St Benedict. In 1098, they acquired a portion of land at Citeaux and founded what was to be the motherhouse of the Cistercian Order. By 1118, the monastery at Citeaux was well enough established to send out monks to create other Cistercian monasteries, and what was to be a Europe-wide spread began. Eventually, the Cistercians would have more than 700 religious houses throughout Europe.
The Cistercians followed the Rule of St Benedict and were known as the White Monks, because of the undyed wool habits they wore. It was impossible for them to live entirely free from the influence of the outside world and lay brothers were engaged in most monasteries; secular men who worked the lands belonging to the abbey and dealt with buying and selling the goods necessary to keep the monastery running.” http://www.suite101.com/content/the-medieval-origins-of-the-cistercian-order-a75209
Tintern Abbey in Wales (see video above) was the first Cistercian house established in Wales. It was part of the first wave of Cistercian houses, supported by the Normans. The second wave, which included Strata Florida (https://sarahwoodbury.com/?p=4863) and Valle Crucis (https://sarahwoodbury.com/?p=4810), were supported by the native Welsh princes and the abbeys, in turn, supported them in their struggle against the Norman conquerors.
The Cistercians were the religious order most sympathetic to the Princes of Wales and Welsh nationalism in the 13th century. http://www.castlewales.com/abbeys.html