Kernels are the core system. Sort of like drivers. Custom Kernels affect stuff like Overclocking for speed, Undervolting the processor to save battery, GPS locks, WiFi range, etc.
They can increase stability, speed, and battery.
The big thing, I think, that most people sought after is the Overclocking/Undervolting (OC/UV). This overclocks the processor while lowering the power usage of the chip at the same time. So you get a faster phone while saving battery. Brilliant. It is my understanding that custom kernels come with normal settings by default. They only ALLOW overclocking abilities using programs like SetCPU and the likes. I think the voltage is lowered on all speeds though, so flashing an UV rom will save you battery even if running 1ghz stock speed. I've seen people go up to 1.4ghz.
It is best to test a rom first for a few days. Figure out what bugs you encounter, if any. Get a feel for the rom so when you flash a new kernel you can see if it will work for you. If you do too much at once and you encounter bugs it will be hard to pinpoint.
I've flashed custom kernels but don't do it often. I know CM roms have their own custom kernels that I believe are flashed back when flashing the rom. This is why CM7 differs from stock in a lot of ways. One example is stock G2x won't work with Wii remotes. The bluetooth stack in the kernel is a custom LG one and doesn't work. CM uses a more common bluetooth stack so you not only gain this functionality but also others.
The list goes on and on. I think most of the custom kernels are BASED off CM7 and are only compatible with CM7. I think a custom kernel on top of a stock LG rom will cause bootloops. So the kernel and rom have to work together.
I am not an expert but I think I've touched the bases. Someone correct me if anything is untrue or misinformative. Technicalities aside, this is the gist of it.
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