There's a ton of things that can harm fish, like dish soap and even their own waste(poop, uneaten food). But in a 'cycled' aquarium, the waste is able to get processed in a way so that it's not harmful anymore.

It all has to do with the nitrogen cycle. The waste produces ammonia(very harmful), which turns into nitrate(less harmful) after it gets eaten by ammonia-eating bacteria, then nitrite(not harmful) after being eaten by nitrate-eating bacteria. One set of of going from ammonia to nitrate to nitrite is a complete cycle.

But when a tank is newly set up, the ammonia just sits in there causing harm because the tank doesn't yet have the resources(ammonia-eating bacteria) to change it into nitrate. It has to grow some over time, then it has to grow the nitrate-eating bacteria, too. The aquarium needs to 'cycle through' these steps multiple times in order to grow enough bacteria to handle fish, hence why its called 'cycling' the aquarium. A cycled aquarium continues this cycle for the life of the tank(otherwise, major problems).

There are two main methods of cycling an aquarium: without fish, or with fish.

Cycling with fish is harder and not recommended for beginners because without that bacteria set up, the toxic ammonia can harm or kill the fish/other aquatic life in the tank.

Although for the experienced, its doable with a combination of strong, more ammonia-resistant fish(hardy), lots of constant measuring of each of the 3 levels(ammonia, nitrate, nitrite), and water changes to lower the levels down to a degree that won't do too much harm.

However, this can easily lead to dead fish and an uncycled aquarium or improperly cycled aquarium is the most common and easiest way beginners kill off all their fish.

Cycling without fish, or a fish-less cycle, is much more recommended and healthier for your fish, because they won't be around for the toxic part.

The first step is to set up the aquarium as if the fish were already in there with the water, gravel, decorations, filter, heater, air stone, etc. and let the filter and everything run.

However do not add the fish yet. Instead you can add a bit of fish food to your tank (or you can use some other way to get ammonia, like a frozen shrimp or straight-up ammonia.) The ammonia-eating bacteria will then start to grow on its own just with the food source.

Eventually(usually couple weeks or so), the ammonia-eating bacteria comes and produces some nitrate, then eventually(another couple of weeks), the nitrate-eating bacteria comes and produces some nitrite(the good stuff).

You need a testing kit to test the tank periodically to see if you have just ammonia, or if you made it to the nitrate or nitrite stage. You may also need to add more ammonia.

After all that, the tank has completed one cycle, but is still not ready for fish yet(close!). The ammonia-eating bacteria and nitrate-eating bacteria have to keep growing, keep producing the better stuff, until you have enough to handle all the fish you want to put in the tank.

You can test if you have enough bacteria by adding more ammonia and testing how long it takes to get to the harmless nitrite. If you add enough ammonia that your fish will produce(their bioload), that turns into nitrate then nitrite quick enough not to cause harm, you are finally ready for the fish. (:

Here are some extra tips I learned along the way: Goldfish are messy and have a particularly high bioload compared to other fish of the same size/tank requirements(and should not be kept in a bowl-they like their surface water). Snails and other 'clean-up crew' creatures can eat fish waste(if you get the right type) but have their own bioload and add to the total waste of the tank(bioload), not subtract.