For a wealthy medieval person, their bed would be quite similar to ours. It would likely be a mattress stuffed with straw or feathers sitting on a wooden frame with slats or rope for the mattress to sit on. The pillows, blankets, and sheets would all be perfectly familiar to a modern person and likely of similar comfort.

For the poor....what bed? A bed is a lot of work to make! There's a lot of wood that has to be carved, a large mattress and pillows to be stuffed, sheets, blankets, etc. Without the benefits of industrialization, you've got a ton of material and time that goes into making a bed. No, the average peasant wouldn't have had a bed at all. They would pile furs and skins up on the floor or a bench (if they had some to spare) and wrap themselves in a blanket to stay warm.