What was the role of monks and monasteries in medieval times? Were they respected?

This is kind of a huge question, and there's obviously going to be a great deal of variation, but I'm going to give it a go. I'm not 100% sure what you're looking for in terms of an answer, so I'm going to write about something along the lines of 'how did the rest of society see monks and monasticism in the Middle Ages?' - if you had something else in mind, then of course feel free to correct me.

In short, yes, monks and monasteries were respected in the medieval period, but in fact I'd go even further and say that monasticism and its central role in society was one of the defining features of the middle ages. The played an unparalleled spiritual role in the minds of medieval people, as they were on the front lines of an eternal spiritual war between heaven and hell. Monks were the archetypal oratores, those who prayed, renouncing all wealth, relationships and even legal personhood in order to devote their lives to prayer for the souls of the faithful. There were the original milites Christi - the soldiers of Christ - fighting Satan and his demons from within the confines of the cloister in order to save mankind from eternal damnation. As a result, they were enormously respected and trusted on the whole within society: people sought to be associated with local monasteries, to have the monks pray for them, to be buried there, to be considered a part of the community. Good Christians donated land and wealth to the monks for the good of their souls and to aid the monks in their pious duties and kings and nobles routinely consulted with great abbots and monastic leaders on matters of importance. Monasticism became so important that by the end of the middle ages (in continental Europe, in any case) it was difficult to find anywhere that was more than a days walk away from some sort of collective religious establishment, whether it be an abbey, priory or collegiate church.

But, of course, this is /r/AskHistorians, so nothing is ever as simple as it looks. Part of the reason we have this picture is that the vast majority of our written sources come from ecclesiastical writers, and the majority of those writers were monastic. So it's only natural that we're presented with a view of monasticism which puts it at the centre of the medieval world, so to speak. Much of the viewpoint I've expressed in the previous paragraph is the product of the tenth and eleventh centuries, which saw huge upheavals in the Western Church. Before and after that, things did and could look different.

In the Early Middle Ages, although monasticism was certainly present in Europe it had nowhere near the prominence that it would come to acquire in later years. Many monasteries were hugely important locally, both socially and economically, but on a wider level had very little influence, either materially or spiritually. This social and economic role is not to be underestimated however: early medieval monasteries such as San Vincenzo al Volturno in Central Italy, Redon in Brittany or St Gall in Swabia (modern Switzerland) all preserve records that show the vast wealth that monasteries could amass in this period and how access to, connections with and control of that wealth was often a focus for local politics. The epitome of this phenomenon are probably the early Irish monasteries, such as Clonmacnoise or Glendalough, which became essentially proto-urban towns, large enough to be politically significant in their own right. This is an exception to the rule however, as in general European monasteries before the central Middle Ages had little independence. Abbacies and control of monasteries were often vested in powerful outsiders, particularly bishops, nobles and kings. This was often seen by the church as important to maintain control of religious sentiment and power, embodied in the local bishop and count (as one can see, for example, in the works of Gregory of Tours). Abbeys were also often founded as a local centre of power by aristocrats (including royalty): a place where the family could be buried and land could be held in trust for the family and controlled through the appointments (usually of family members) to the abbacy. It eventually got to the point that, in the tenth century, most monasteries were openly controlled by laymen, either as lay abbots (common in the lands that would become France; the Robertian family, who would become kings of France in 984, based much of their power on control of a string of lay abbacies in the Loire valley and around Paris) or as advocates (lay persons appointed to deal with an abbey's temporal and legal affairs, since men of God shouldn't deal with such things; a position much more common in German-speaking areas than further west).

Just as this trend came to a head, however, things began to change. The process known to historians as 'Monastic Reform' was actually a series of independent, though linked, reforms of monasteries based on the idea that they should be independent of secular control, or even the control of local bishops: abbots should be elected by their monks, not appointed by outsiders and the property of the monastery belonged to that monastery's patron saint, and should therefore never be alienated or controlled by lay persons. From several notable centres in the tenth century (Cluny in France, Gorze in Germany and Ely and Ramsay in England), these ideas gained huge currency and popularity by the middle of the eleventh century, and were the basis of what would come to be known as the 'Papal Reform Movement' (sometimes called the 'Gregorian Reform'). It's notable that many of the biggest intellectual leaders of this wider movement for church reform were monks themselves and many of the movement's aims could be classified as making the church as a whole more 'monkish', such as removing secular appointments to bishoprics and forbidding clerical marriage. The eleventh century reform movements were much broader than just monasticism, but they held up the monk as a spiritual ideal for the clergy and for society at large. Monks produced huge amounts of history, hagiography and theology which all promoted the ideal of monks which I outlined in the opening paragraph, and people, by and large, seem to have bought into it. Bishops, archbishops and popes were all much more likely in the eleventh and twelfth centuries to have started their careers as monks, which only helped to spread the ideas and ideology of monasticism. New monasteries were founded across Europe in huge numbers, and older, crumbling ones were reformed and re-founded to great fanfare. People, not only aristocrats but ordinary farmers and peasants, donated land and wealth to monasteries on a scale never before or again seen and the most famous monasteries became veritable mini-cities: at its peak, Cluny had 300 monks and thousands of support staff. New monastic orders abounded in the twelfth century too, as people sought to find new and purer monastic ways of living: here, the Cistercians are notable for their encouragement of commoners and peasants to join the monastic life as conversi (sometimes translated as 'lay brothers'), who joined the monastic orders but spent the majority of their time working for the monastery in its fields, rather than praying in its chapels. The economic effect of the monastic boom is important to note too: there was a huge investment by monasteries in building new churches and support buildings, as well as considerable investment in the productive capacities of the land they owned, particularly converting un- or under-productive land to be used to produce marketable goods (the Cistercian order are the most famous for this, but they were by no means alone in their practices). The social role played by the monastery locally continued unabated, where it remained the biggest local landholder, giving patronage, aid and promotion to those people or families who were close to it. The late tenth to the early thirteenth centuries can be characterized as the golden age of Western monasticism, where my opening description is most fitting.

In the later middle ages, this picture begins to dim a little. I won't speak as much about it, as I know a lot less about later medieval monasticism (hopefully a medievalist with a slightly later area of expertise can chime in?) but on the whole there's evidence that people were becoming more and more dissatisfied with traditional monasticism. This can be seen in the rise of more specialised or personalised devotional practices among the laity (particularly the urban classes) and the rise of new religious movements which emphasised poverty and apostolic living to a much greater degree than pampered monks did (both orthodox movements like the mendicants and heretical movements). The respect for monks and monasticism never disappeared entirely during the middle ages, however, and that golden ages of central medieval monasticism gave us some of the ideas which continue to define Western Christianity to this day.