Anaerobic digestion uses microorganisms to break down waste constituents. Anaerobic means no air. Composting uses microorganisms too, but that is an aerobic process. Anaerobic digestion takes place in a closed vessel or location where little or no air can get in. It is very similar to what happens when dead plants decay in swamps. As in swamps, a burnable gas can form in the digester.
The waste to be processed in anaerobic digestion is called feedstock. Feedstock can consist of food, yard waste, and potentially even biological waste if that is permitted by regulators and a hazard analysis. We are unaware of things like used bandages being treated this way, but it is feasible.
Paper is probably not an ideal feed for AD as it is slow to decay. Cloth and fabric are better suited than paper. Sewage sludge and animal manure are ideal feeds. If the waste stream has a inappropriate waste such as plastic in it, a screening preprocessing may be needed to remove it for recycling or landfilling.
The gas that evolves from AD is analogous to swamp gas. It is largely carbon dioxide and methane, and does have combustion value. Large operations may have a method to recover this energy for cogeneration of electricity. The “bottoms” from the process are called digestate. It can be used as a fertilizer or disposed of by landfill or composting. Highly optimized AD processes tweak the carbon to nitrogen ratio in the feed.
Types of digesters go by names in the industry: upflow anaerobic sludge blankets, expanded granular sludge beds, anaerobic clarigesters, anaerobic filters, and hybrid biological reactors. A diagram of an anaerobic process is here: https://www.fao.org/3/t0541e/T0541E02.GIF