WinDbg is the GUI version of the Debugging Tools for Windows, the free (as in beer) debugger package released by Microsoft to facilitate the composition of software on Microsoft Windows. It has been continually refined since Windows NT and has, despite its primitive interface, accumulated an incredibly powerful mess of features.
WinDbg is ugly compared to the debugger in visual Studio, but it is also significantly more powerful. Its !analyze extension automatically diagnoses the causes of the sweeping majority of program or operating system crashes, down to the exact line at which memory corruption or pointer mishandling occurs. For more complex tasks, a single command can dump entire Standard Template Library data structures, or extract data from custom classes and, if they are shaped vaguely like a linked list, automatically follow pointers to give structure information.
WinDbg has been designed primarily for the developers of the Windows operating system itself, and therefore has a lot of special commands designed largely for use inside of Microsoft. Some of these are generally useful enough to ship externally in the public downloadable package anyway. Some are named based on inside jokes; the most notable of these is the extension !itoldyouso, to give brief information on why the debugger is or is not using a certain set of symbols to decompile a program, written after hundreds of messages complaining about the debugger complaining about mismatched symbols when the symbols were, indeed, mismatched.
WinDbg can be remotely controlled by another instance of WinDbg somewhere across the network, a critical feature when the test computer a crash occured on is nowhere near the programmer who wrote the bug. This is especially important in overly large companies like Microsoft, where the people involved may not be on the same continent.
Like all debuggers, WinDbg will not actually remove bugs in your code for you.