'Cycling' is the buzz word in the aquarium-keeping hobby, and rightfully so. But just what does it mean? Well, let's check out some definitions of 'cycling'.
With how pervasive cycling is, it would seem obvious that the process is well established by now. One quick Google search and I think we all know that is not the case. There are so many guides out there, and they all seem to differ. What gives? Well, one reason I can say is, we do not even have a consistent definition of what 'cycling' is supposed to be.
With that in mind, I will go through the definitions that exist in this hobby. I will also describe some very niche definitions that some people use, just in case anyone is curious. These definitions often fall into disuse, either because they are simply wrong, or describe/include processes that the majority of us can at least agree is distinct from the 'cycling' process.
Aerobic - something that respires ('breaths') using oxygen.
Anaerobic - something that can respire (breath) without oxygen.
Ammonia - the product of fish waste, decomposition, and just general ammonification. Toxic to both freshwater and marine fish.
Ammonification - conversion of organic nitrogen into ammonium (ammonia). This is really as much as you need to know.
Bioload - The 'biological' limit that an aquarium can hold, generally measured as a combination of numbers of fish. That or effectively, how much food is ultimately added to an aquarium over a day/week/etc.
Biomedia - Anything that can be colonized by (beneficial) microorganisms. Technically anything with a surface, but probably better considered to be products specially designed or designated for settlement of beneficial microorganisms (bioballs, ceramic noodles, live rock, lava rock, etc.).
Cycling - everyone defines this differently, but I define it as 'to establish enough nitrifiers in an aquarium to fully handle ammonia produced by a full bioload'.
Microorganism - small living things. Include bacteria, but also other living beings, such as algae, fungi, and so on. There is some debate amongst scientists about what can exactly be classified as a microorganism. Mostly irrelevant to the aquarium-keeping hobby, so don't worry too much about it.
Nitrification - The process in which ammonia is oxidized to nitrite then nitrate. Do not worry about what 'oxidized' means, you can use the term 'converted' instead. Not exactly important, but may be of interest: ammonia can actually be directly converted to nitrate, by microorganisms such as Nitrospira inopinata.
Nitrifier - A microorganism that either converts ammonia to nitrite, or nitrite to nitrate (or ammonia to nitrate directly, see nitrification). Common genera ('groups') of nitrifiers are Nitrosomonas, Nitrosospira, and Nitrospira. In marine environments, certain nitrifying Thermoproteota species are also common, specifically from the genera Nitrosopumilus and Nitrososphaera.
Nitrate - The final product of nitrification. Nitrate can be consumed by certain organisms, including microorganisms in anaerobic processes, or by plants and algae. Nitrate toxicity varies greatly depending on live stock species, but generally not as much as ammonia or nitrite.
Nitrite - The product of ammonia oxidation, which is then oxidized to nitrate. Nitrite is very toxic to freshwater fish, but needs to be at very high concentrations to be toxic to marine fish. Toxicity varies between marine fish species, but probably around 100+ppm is enough to be lethal for some species, and higher for others. I would suggest nonetheless keeping nitrite below 25ppm for marine systems, as even if not lethal, higher concentrations of nitrite can still cause disease.
Alright, now for the part you wanted! Definitions of 'cycling'.
1. Building up enough nitrification capacity for an aquarium to handle a full stock. This is the most common definition, even if a lot of guides do not explicitly say it. It is very inherent in the way a lot of guides proposed how to determine an aquarium is cycled, and you often see it as well - any guide that says an aquarium should be able to handle at least 1ppm a day (2ppm and 4ppm is also common, 8ppm is proposed here and there) is also saying that the aquarium should be able to handle all the ammonia produced by a fully stocked tank each day. This is of course advantageous, because you are able to fully stock a tank if you want or need to. It also ensures there is no ammonia spike with each subsequent addition of live stock. Once we realized how effectively, permanently, and easily we can keep ammonia in check (via nitrification), well why not!
The important thing to know is, for typical tanks, cycling will occur eventually even if one does not intend for it to happen. As fish releases waste, that is converted to ammonia, and that is used up by nitrifiers, allowing them to grow and multiple. If someone ever told you about them buying fish again and again and again and suddenly one day deaths no longer occur... well they probably killed all their fish initially with ammonia and/or nitrite spikes, until eventually it stopped being a problem as their aquarium eventually become cycled.
2. Ensuring ammonia and nitrite is kept at zero. Ultimately the reason to cycle a tank is to keep our live stock safe, specifically from the effects of ammonia (and nitrite). So if ammonia and nitrite can be kept at zero, then theoretically that's fine, no matter how it is done. While nitrification is one method to prevent ammonia (and nitrite) build up, that's not the only way ammonia is biologically consumed. Plants also consume ammonia, and so some define cycling as simply the process in which a tank is established to consume all ammonia produced - no matter by nitrification or by using plants. Other organisms also do consume ammonia - a lot of microorganisms use it as a nitrogen source, much as plants do. Theoretically the way they do it is fine - after all, zero ammonia is zero ammonia, right? This is where there is quite a bit of debate, specifically surrounding the sustainability of the method. What if the plants die? Suddenly that method of ammonia consumption is gone. Meanwhile, consumption of ammonia as a nitrogen source by heterotrophic bacteria can lead to bacterial blooms that can eventually suffocate the tank by depleting oxygen. The only organism that sustains consumption of ammonia without needing to grow in population, and are pretty hardy, are nitrifiers. This definition is not uncommonly adopted, primarily because methods such as fishless cycling using fish food has to rely on it.
These are the two main definitions of 'cycling' and are widely accepted. Now, as promised, are definitions of 'cycling' that has arose over time and fell into disuse as well. They should not be considered, but I am including them here so that you will not be surprised if you ever come across it.
3. Ensuring nitrification is simply taking place in the tank. You most likely have stumbled upon that typical graph where ammonia spikes and drops, then nitrite spikes and drops, then nitrate spikes. And then the accompanying guide will say, congratulations, your tank is cycled! Here, the definition of 'cycled' simply means there is some level of nitrification going on. But how much? That's the issue. Say it took a month for ammonia to drop from 2ppm or whatever to zero. That means that on average, the nitrification rate was only 0.066ppm ammonia a day. Of course, because nitrifiers would probably have been reproducing during that time, you likely had a higher nitrification rate towards the end of that month - so let say it even reached 0.1ppm. That's not enough in most cases. Say a betta in a 10 gallon tank, producing even just 0.5ppm ammonia a day - you are missing the mark by 0.4ppm, and thus now you have a situation of a fish-in cycle. This only works with larger tanks and slow stocking. If ten fish in a 100 gallon produces 1ppm ammonia a day (just an example), then one fish only produces 0.1ppm... and so yes, if one adds one of these fish to their 100 gallon tank that can only handle 0.1ppm a day, then that one fish is safe. Note: this is now widely accepted as an extremely bad way to define cycling, especially for new fishkeepers who can 'cycle' their aquarium(s) this way, then measure ammonia/nitrite down the line, see a lot present, and then panic having no idea where they went wrong.
4. Establishing a diversity of beneficial microorganisms. Nitrifiers are not necessarily the only beneficial microorganism in an aquarium. Some consumes other types of waste products, others can compete against pathogens or improve the health of live stock somehow. Quite a few form symbiotic relationships with aquatic plants, promoting their health too. Some define 'cycling' as the process of ensuring the establishment of ALL (or at least the ones relevant to one's tank, i.e. planted versus non-planted and so on) of these diverse beneficial microorganisms. This definition of cycling is used by only a very small portion of the community, due to two issues. First - it is hard to define what types of beneficial microorganisms are important right off the bat and what are not, and what likely comes along with introduction of live stock anyways. Second - it steers rather far away from what 'cycling' is originally about. After all, the term 'cycling' is derived from the 'nitrogen cycle', whereby the metabolism of ammonia (and nitrite, but also nitrate) occurs. Clearly ensuring the tank's microbial health is robust can be important, but that is far more encompassing than just the handling of nitrogenous substrates. Thus, the majority, but not all, aquarists do not consider this as a valid definition of 'cycling'.
5. Establishing the aquarium. This was promoted for a time, but fell into disuse quite rapidly - even more so than (4). This is understandable, given the question - what is 'establishing the aquarium'? A lot of definitions and descriptions are thrown about - from the necessity that algae must grow then die off, to adding all manners of substrates (soil, sand, rotting plant matter, etc.) to a tank and letting chaos birth a healthy ecosystem. While many of these processes do have their benefits, the realization is, they can and do stand separately and thus really should not be included in the definition of 'cycling'. Thus, the majority, but not all, aquarists do not consider this as a valid definition of 'cycling'.
6. Letting the aquarium run for [x] amount of time. Basically no matter what the setup, what is added or what is not, some simply says cycling is to let the tank run for some time without touching or doing anything to it at all - generally a month, though three weeks, and three months are also common amounts of time thrown around. This is certainly problematic, least bit because of the diversity of aquariums out there. You can imagine an entirely empty tank (devoid of substrate, etc.) with water just added and ran for a month. Well, what does that specifically establish? Nothing, really. So, this has been for a long time widely accepted as an invalid definition of cycling.
And that's it! So now you know why it can be confusing at time when talking about what cycling. It is unfortunate that there had been so many attempts to diverge on the definition and make it confusing. Sometimes it is a matter of slight differences in targets (1ppm versus 4ppm ammonia oxidation/day rate for example). Sometimes it is trying to include other, potentially beneficial processes in the definition of 'cycling'.
Hope this has clarified a lot of things for you!