The nitrogen cycle are the processes that convert ammonia (fish waste) into less harmful nitrate which you typically remove through your regular water changes. This is mainly done through microbes that eat the ammonia and excrete nitrate via an intermediate nitrite stage. These microbes live in places where the is surface area to grow on and a good flow of oxygenated water. In aquariums this is mostly going to be in your filtration, and they grow on those beads in that cartridge in your photo and the sponge. To a lesser extent they will also grow on your substrate, the glass your aquarium, your aquascape, anything with surface area to grow on.
But aquariums dont come with these microbes built it, you have to grow them. "Cycling" an aquarium is the process of growing enough of these microbes to consume the ammonia your fish produces and turn it all into nitrate. This typically takes a couple of months, and until you are cycled toxic waste (ammonia and nitrite) will build up in the water between water changes, and when you are cycled you will see zero ammonia and nitrite, and less harmful nitrate will build up between water changes.
In short, as you already have a fish you should be doing what's called a "fish in cycle". Essentially this involves frequent testing of water parameters and changing water before ammonia and/ or nitrite get to toxic levels. In newly set up aquariums this might be a daily water change, but as your cycle establishes and those microbes grow, you will see less ammonia and nitrite and need less frequent water changes, and when you consistently see zero ammonia + nitrite you are cycled and can settle into a weekly/ fortnightly water change routine.
You will need a reliable test kit that tests for pH, ammonia, nitrite and nitrate. Avoid test strips as they arent always reliable. API Freshwater Master Test Kit is commonly used. It covers what you need, is reliable enough for what you are using it for, and fairly easy to use.
Here is a more detailed guide on how to do a fish in cycle. You are already part way through. Have a read through, but you are going to be at Stage 3.
STAGE 1 – SET UP AND PREPARATION. Set up the aquarium, stand, filtration, lighting, heater if required and any other equipment. Fill the tank and run for 24 hours to ensure there are no leaks and that all the equipment functions correctly. Drain the tank. Aquascape including substrate...
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