I am unable to follow where exactly
  RuntimeException should be used 
That's probably because you are looking at an argument, i.e. people are disagreeing about exactly this point.
and
  how it is different from normal
  Exceptions and its subclasses.
Very simple: All subclasses of Exception (except for RuntimeException and its subclasses) are checked i.e. the compiler will reject the code unelss you catch or declare them in the method signature. However, subclasses of RuntimeException are unchecked.
Googling gave me a complex answer,
  that is, it should be used to deal
  with programming logic errors and
  should be thrown when no Exception
  should normally occur, such as in the
  default block of switch-case
  construct.
This is the conventional wisdom, which says that for everything that a program can usefully deal with, you should use checked exceptions because then the compiler will force you to deal with them. Conversely, programs can typically not deal usefully with programmer errors, thus they don't have to be checked. This is how the Java Standard API uses RuntimeException.
The discussion you linked to is sparked by the view of some people (this includes me) who think that checked exceptions lead to bad code and should therefore not be used. Since you can't disable exception checking in the compiler, the only way to do this is to use only RuntimeException and its subclasses.
One observation that IMO supports this view is that the conventional wisdom of "use unchecked exceptions only for programmer error" is in fact mainly a rationalization of backwards-reasoning: there is no code safety reason why the compiler should not force you to deal with programmer errors. However, something like NullPointerException and ArrayIndexOutOfBoundsException can crop up almost anywhere, and if those were checked, nobody would ever want to program in Java. Thus, the language designers had to make a, huh, exception for those, and make them unchecked. To explain this, they came up with the "unchecked exceptions are for programmer errors" story.