Everyone today, except for the truly homeless, would be considered rich by medieval standards, because of everything they own or have access to, from electricity to plumbing to furnaces/air conditioning to cars to TV to refrigerators to phones. But as I discussed earlier, there were plenty of people considered wealthy in the Middle Ages, and gradations of wealth down to the destitute.
Too often today one hears suggestions that those poor in the modern world somehow chose not to have enough money, being lazy and shiftless people who prefer to survive with handouts. This isn't particularly true now, but in the twelfth and thirteenth centuries there really were people who deliberately chose to be poor and to live by handouts. They were considered holy.
In the New Testament, Jesus is described as wandering from place to place, without having any sort of permanent home, preaching as he went and accepting food and hospitality. The apostles followed him, and indeed the Bible notes that Jesus said that others should do the same. Now I personally (but I am not a minister!) think a person can be a good Christian without wandering barefoot around the Sea of Galilee, but the Bible is remarkably explicit on this point.
Following this example, there were a number of people in the High Middle Ages who deliberately gave up all material possessions and comforts, either to live a life of prayer as hermits separate from the world, or, increasingly commonly, to wander around and preach. Monks had been around in the West since the fifth century, people who lived in groups and who shared all their possessions (following the Acts of the Apostles) while leading deliberately simple lives. Hermits were rarer but still found, men who lived a solitary life of simplicity and prayer, far from human habitation, although they had to be close enough to humans for people to come and make offerings.
But wandering preachers were different. They were disruptive and meant to be. They considered themselves to be following the New Testament, including all the indications that Jesus was considered threatening by the well-established religious leaders of his time. For that matter, the secular governors (the Romans in first-century Palestine) considered him disruptive enough to execute him.
With any self-proclaimed holy disruptive person, the question is always are they genuinely inspired by God or genuinely a crackpot. The bishops of the twelfth century tended toward the latter explanation. Wandering preachers would come into town, preaching about salvation and damnation, and leave with a new group of followers, often disproportionately women. Now a lot of these new followers would quietly return home within a couple days, realizing that not having much to eat and sleeping out in the rain was not a comfortable lifestyle in spite of the promised salvation. But some would stick it out.
The bishops wanted these preachers to settle down and become monks, and especially they wanted to make sure they got a good religious education, because it was unclear what their message of salvation and damnation was based on. Some of these preachers spent much of their lives skating along the edge of heresy, promising to settle down and be monks and then heading off cross-country as soon as the bishop's back was turned.
But absolute poverty and wandering preaching became officially accepted by the organized church with the Franciscans (founded by Saint Francis), recognized by the pope in 1215. They took voluntary poverty to a new level, living entirely by begging, refusing to touch money, not even saving an apple over from one day to the next--eat it if you're hungry, if not give it to someone who is. People were stunned by their holiness and similarity to Christ and the original apostles, a connection made explicit when Francis received the stigmata. The absolute poverty he adopted started being modified shortly after his death—it really was an almost impossible standard. Later in the thirteenth century, those Franciscans who said they were sticking with Francis's original message got embroiled in an apocalyptic heresy, which sort of ended that branch, but the ideal of holy voluntary poverty long persisted.