Nitrogen cycle
The biological process where bacteria in a tank's filter convert toxic ammonia → nitrite → less-toxic nitrate. A tank is 'cycled' when this conversion happens fast enough to keep ammonia and nitrite at zero.
Fish waste, uneaten food, and decaying plant matter all release ammonia, which is acutely toxic to fish even at low concentrations. In an established aquarium, two groups of bacteria handle this. The first group (Nitrosomonas) eats ammonia and produces nitrite — also toxic. The second group (Nitrobacter / Nitrospira) eats nitrite and produces nitrate, which is far less toxic and is removed by water changes or live plants.
A new tank has neither bacteria population yet. Adding fish before the cycle is established exposes them to ammonia and nitrite spikes. The 'cycling' process is what builds up these bacterial colonies — typically 4-6 weeks for a fishless cycle, longer for fish-in cycling.
Most beginner trouble in the hobby traces back to skipping or shortcutting the cycle. Cycling first, stocking second.
Beginner-safe rule
Conservative beginner-safe rule: do a fishless cycle by dosing pure ammonia to ~2 ppm and waiting until both ammonia and nitrite drop to 0 within 24 hours of dosing. That usually takes 3-6 weeks. Once that happens, the tank is ready for fish.