By modern standards, everyone was dirt poor in the Middle Ages. Even the wealthiest lacked the "things" that everyone but the truly homeless now take for granted: electricity, indoor plumbing, central heating and air conditioning, telephones, washing machines, automobiles. There were no TVs or movies, no radios or CD players, no subways or buses, no supermarkets.
Nonetheless, there were always gradations of wealth, with some people considered very rich. Their wealth was based more on the power to command than on their bank accounts, but those considered rich did have much nicer clothing (and more outfits), better houses and furniture, and on average better food than those considered poor. Everyone, as is the case now, would have liked to be rich.
During the early Middle Ages especially, wealth was rare. During this time holiness and wealth were often equated. God was referred to as the King of Kings, and He was worshipped in churches that might, by later standards, be cramped and dark, but on the inside there was plenty of precious metal on the vessels and silken hangings on the walls. Bishops lived like princes, which was considered entirely appropriate.
It was clearly understood by everyone that helping the poor was a good Christian duty. There were however so many poor people that those with wealth essentially gave up. When most of the population was trying to scratch out a subsistence living, it was hard to know where to start.
But with the economic upturn of the twelfth century, it actually became possible to climb out of poverty. Merchants could make enough money to rival the nobility in their possessions and lifestyle. Some peasants became wealthy enough to hire other peasants to do their farm chores for them. Young men tired of trying to farm unsuitable ground could move to the rapidly growing cities to "make their fortunes" (or at least get a little ahead).
At this time, holiness became defined not with wealth but with poverty, in particular voluntary poverty. Monks had always lived more simply than bishops, but now the new monastic orders sought to give up all personal possessions and live austere lives: vegetarian diets, undecorated churches, hard manual labor. The King of Kings became instead the itinerant preacher of the New Testament, begging for a living. The new monastic orders liked to point out that they were much humbler and poorer than bishops or older monasteries, and hence holier, even though (as they recognized themselves) it was hard to avoid being proud of how humble they were.
Older monastic orders, although not adopting poverty quite so radically, still sought to assure that their monks had only collective property, not individual property. All of them continued gifts to the poor. For example, any food left after the monks had eaten was distributed to beggars. Some monasteries had a group of official poor people living outside the gates, to whom such alms were given and whose feet would be washed on Maundy Thursday.
Today no one voluntarily chooses to be poor (though you might think so, based on the punitive nature of some supposedly anti-poverty programs). But voluntary poverty was greatly admired in the High Middle Ages (see more here). The most extreme example was the thirteenth-century Saint Francis (whose name the current pope took), who urged his followers to live by wandering and begging, not even keeping an apple overnight to eat the next day--eat it if you're hungry, but otherwise give it away. The Franciscan order, unable to maintain this rigor after his death, ended up living essentially like austere monks.
Poverty as a real threat, not just something to be avoided--or chosen--came back in the fourteenth century. Europe was becoming over-populated, at the same time as the climate was worsening. Famines broke out on a scale not seen for several centuries, and desperate bands of hungry people might roam the countryside, looking for food.
It was at this time that the distinction between "deserving" and "undeserving" poor was first made. Now donors wanted to know if the poor they were asked to help really deserved it, were honest, hard-working people who had suffered disasters, or if they were shiftless bums (and probably drunks) who got what was coming to them. This is a distinction the modern world still makes.
The fourteenth century was also the time of the Black Death and the outbreak of the Hundred Years War. It was also the beginning of the Renaissance. A bad time all around.