So as mentioned in the page on breeding, after you breed with a mate, the resulting egg goes into your Egg List.
Here you have 8 available slots to hold your eggs until they’re ready (or you are) to hatch them into your pond. Eggs seem to take a different amount of time to be ready to hatch based on the patterns of their parents.
It’s important to keep the numbers of eggs in your list below maximum, because if you have the Egg List full, you will not be able to breed.
If you have space available in your “My Koi” you can go ahead and hatch some of the ready eggs into your pond. When they hatch, you will discover what color and pattern they are, and that combination is immediately registered in your collection. After that, you can decide whether you want to trade, release or ascend them.