1) Ammonia is produced both by fish and by microorganisms that decompose organic matter. The decomposer microorganisms are beneficial in that they are part of the group that is removing fish feces, fallen food, dead leaves etc. This group includes heterotrophic bacteria, fungi and other organisms. If these beneficial organisms (including bacteria) were not present the waste would not get broken down into plant fertilizer.
2) Ammonia is removed by microorganisms that are still being studied for ID, but seem to be in a different group of organisms called Thaumarchaeota. These are not the same as bacteria, but for aquarium keeping purposes are still referred to as beneficial bacteria. They remove ammonia, and oxidize it into nitrite. Would you rather have ammonia in the tank?
PLOS ONE: Aquarium Nitrification Revisited: Thaumarchaeota Are the Dominant Ammonia Oxidizers in Freshwater Aquarium Biofilters
3) The nitrite from AOA (Ammonia oxidizing Aechaea) is removed by organisms related to Nitrospira species of bacteria. These also are beneficial bacteria. They oxidize the nitrite into nitrate. Would you rather have nitrite in the water?
In all three of these there certainly may be more than one organism that are doing the work. But in common terms the whole group is referred to as beneficial bacteria (even if many are not bacteria).
The organisms in points 2 and 3 grow stuck to surfaces such as biological media, sponges, substrate particles, and the surfaces of all the things in the tank such as driftwood and ceramic merpeople. They grow in a complex structure called bio film with many species of organisms. The decomposing organisms may be found with their food, the organic matter that is commonly found at the bottom of the tank, in all the nooks and crannies. Even if you did a very good job of removing as much of this debris as you can, as often as you can there will still be food for these organisms, and you will not remove them all from the tank. If you did (perhaps by sterilizing everything) more microorganisms would land in the tank and begin growing. (These are the organisms that can cloud the water, they can grow so fast). There are plenty of them living in the filter media, too, so not matter how well you clean (vacuuming out the organisms with the things they are eating) there is a good supply of them ready to eat the new waste.
The term "Cycled" has several meanings in aquarium terms, but the most common use is in the 'Nitrogen Cycle'. This is the growth of organisms in meanings 2 and 3 that remove ammonia and nitrite and turn these into nitrate. It takes time to grow these organisms, and this time is called 'Cycling the tank'.
The growth of organisms in meaning 1 above is not really called cycling, though these organisms also increase in population along with the nitrifying organisms. When a fish species is said to require an established tank, the overall concept is that the species that decompose stuff and the nitrifying group are well established and are keeping the water parameters in good, stable condition.
The opposite of 'beneficial bacteria' are the disease causing organisms. These are the 'bad bacteria' (and virus, fungi etc.)