What: When Windows starts, it also loads in loads of other software that you have installed. "Safe Mode" only loads the barest minimum of the software and device drivers. Usually it is invoked by pressing F8 during the boot sequence.
Why: It enables you to "get into" Windows when a software or driver conflict is preventing this, giving you the chance to try to diagnose the problem.
When: You'll only need to use it if advised by someone else or if you fancy trying to diagnose a problem yourself.
As an example, my nVidia (graphics) drivers recently started playing up - soon after logging on, my monitor went blank. By going into safe mode "with networking" I was able to download and install a new driver to replace the corrupted one, fixing the problem. (Because graphics drivers are not loaded in Safe Mode the problem did not occur).