Kernel debugging is performed on two systems because there is only one kernel; if the kernel is at a breakpoint, no applications can be running on the system. One system runs the code that is being debugged, and another runs the debugger. Additionally, the OS must be configured to allow for kernel debugging, and you must connect the two machines.
It is possible to run a kernel debugger on the same system as the code being debugged, but it is very uncommon. A program called SoftICE used to provide this functionality, but it has not been supported since early 2007. No vendor currently offers a product with this functionality.
There are different software packages for user-mode debugging and kernel debugging. WinDbg is currently the only popular tool that supports kernel debugging. OllyDbg is the most popular debugger for malware analysts, but it does not support kernel debugging. WinDbg supports user-mode debugging as well, and IDA Pro has a built-in debugger, but these do not offer the same features or ease of use as OllyDbg.