Medieval people threw away far less than we do, for the simple reason that they had far less "stuff" to begin with (as I discussed in an earlier post). In these days, when many Americans fill a 30-gallon drum with throwaways every week, it may be hard to realize how recent are today's landfills.
Throwing things away really started in the late nineteenth century, when food started coming in containers. If you have a tin can or a glass jar that you don't need any more, you throw it away (or, one hopes, these days you put it in the recycling bin). With the spread of plastics during the twentieth century, there was more and more to throw away. Food scraps, broken toys, old newspapers, empty bottles, electronics that have stopped working, styrofoam cups, cracked plates, beat-up cardboard boxes, worn-out tires, stained clothing, unused carpeting from the remodel, off it all goes.
Medieval people certainly generated food scraps, but they would not be thrown away. For one thing, one really did not leave food on one's plate, because there was not today's abundance of food. At a castle or well-to-do monastery, food that was put on the table (in serving dishes) but not consumed would be passed on to the poor. Cooking scraps, vegetables that had gone off, egg shells, cheese rinds, and bits of fat would often be fed to a pig (probably not too good for the pig, but that's a separate issue). If there was no pig, they could just be composted or buried.
Bones and shells were harder to dispose of. Bones could be used to make glue, and shells could be ground and mixed with chicken feed, but often these hard substances were just tossed into a pit or a pile. Such midden heaps are of great interest to archaeologists, because they can determine what kind of animals medieval people were eating from the bones that are left (beef? pork? deer?). The midden heaps would also get a certain amount of other things (like broken crockery) that no one wanted and were not easy to recycle. Sometimes other things (rings, coins) would end up in the midden heap by accident.
Clothing would not be thrown away. Fine clothing that was no longer wanted would be passed down the social ladder. Peasants wore their clothing until it was literally rags. Rags were very useful in an era without facial tissues, toilet paper, paper towels, or feminine hygiene items. Once paper came into common use, rags were also used to make paper.
Rags continued to be valued until the nineteenth century. If you've read any Charles Dickens, you may recall that very poor people could get a little money collecting and selling rags (the way today someone might collect and return bottles).
These days the "sanitary landfill" (what an euphemism) can be enormous. There's one near Chicago big enough to have a ski slope. They are supposed to have "liners" to keep the trash and its effluents out of the water supply; many have doubts. Sometimes landfills catch fire and burn, literally, for years, because of all the chemicals and petroleum products in them, which of course were unknown in medieval times. We might mock medieval people because they did not have the minty freshness of people who have daily sudsy showers, but they would mock us for our belief that we can just throw things away and never worry about them again.