Short explanation: a kernel is the communication interface between software and hardware. If you haven't one, you can't use your phone.
There are custom kernels and stock kernels. Stock kernels are by SONY and they are not "something special". Custom Kernels are by devs(example from the xda forum).
With custom kernels you can for example overclock your cpu. Also you can save a bit of your battery life with some settings.
But for flash a custom kernel, you need to make a TA backup(to keep warranty), then unlock your bootloader.
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