What is the explorer.exe equivalent in OS X?
This is a very wide reaching question, but to answer the central theme, the rough equivalent of explorer.exe on Windows is the Finder application.  Finder is the only application you cannot remove from your dock, mainly because it is the finder that runs the dock etc, much like explorer.exe runs the taskbars etc in Windows (I think I got that right for windows)

You can kill the finder, and it will automatically re-spawn.  The equivalent of Task Manager would be Activity Monitor as a graphical app, and of course you have a fully featured Unix command line environment in Terminal, which allows you to interact with processes using all sorts of commands, for example:

ps -ef (lists active processes)

kill -9 243 (send interupt request 9, or terminate with extreme prejudice to the process number 243)

There is no Registry in OS X, which is nice because that was a horrible idea (in my opinion), but settings and preferences etc are commonly stored in .plist files that reside in either the system or an individual users Library folder, which I suppose could be analogous to the windows\system or user\application data folders in Windows.