The nitrogen cycle is the process where colonies of beneficial bacteria in your filter convert toxic fish waste (ammonia) first into nitrite, then into far less harmful nitrate — it is the single most important thing to understand in fishkeeping.
Fish constantly produce ammonia, which is toxic even in tiny amounts. A mature filter grows two families of bacteria that turn ammonia → nitrite → nitrate; you then export the nitrate with water changes. "Cycling a tank" means growing those bacteria before you rely on them.
Almost every beginner disaster — cloudy water, sudden deaths, "new tank syndrome" — traces back to a tank that was not cycled. Getting this right first is the whole "get the water right" idea.