More than once I've heard "take care of the water and the water takes care of the Koi." That sure sounds good, but truly understanding the Nitrogen Cycle had baffled me until I took the K.O.I. Certified Koi Keeper course.
I can tell you I read and read and read trying to figure it out, but it just never made complete sense, perhaps it was overthinking, kind of like when someone can't swim and they try too hard and sink like a rock! Perhaps, the problem is that there are 2 things that sound almost identical, but are in reality VERY different chemicals - Nitrite and Nitrate.
To keep it plain and simple, the start of the cycle is Koi releasing ammonia into the water (humans produce C02 when they breathe, but Koi produce ammonia when they breathe). The pond pump circulates the ammonia into the filter, where bacteria devour the ammonia (you can think of a filter as just a home for beneficial bacteria!). That doesn't really help the Koi, cause the output of these bacteria is nitrite - which is even more toxic to Koi than ammonia! But, it turns out there is a 2nd colony of bacteria in the filter that start feeding on the nitrites. The good news is that the output of the 2nd colony of bacteria is nitrate, and nitrates are not harmful to our Koi in low numbers Nitrates can either be used by plants as fertilizer, or can be removed with regular water changes. If you think of plants as food for the Koi (well, I hope that you protect your expensive Water Lilies, so the Koi don't eat them!), then you can see that there is a circle - and it's called the nitrogen cycle - as pictured above.
There are many factors that influence this process, temperatures, oxygen, fish load, etc.
Testing your water will be key to tell you that your filter is working and all is well. If the filter is new or becomes damaged, high levels of ammonia or nitrite can hurt or even kill your Koi. So test your water! And consider taking K.O.I. #201 - Water Quality - which will describe the nitrogen cycle in more detail, and tell you how best you can prevent and treat water quality issues, like ammonia and nitrite.