I am trying to (intuitively) understand the two terms "decidability" and "verifiability".

I have done a reasonable amount of searching and going through the various texts I can put my hands on. However, their intuitive understanding seems to escape me, specially for the second one.

Out of the many definitions found, the following one found in [this page][1], clearly explained decidability to me.

> A language is called decidable if there exists a method - any method
> at all - to determine whether a given word belongs to that language or
> not.

However, I fail to find a parallel definition for verifiability.

In the [Theory of Computation book by Sipser][2], we find,

> P = the class of languages for which membership can be *decided*
> quickly. 
> 
> NP = the class of languages for which membership can be
> *verified* quickly.

In light of the above, I want to understand verifiability. 

Please provide as many examples as you can, at one moment, I try catch the meaning, in the next one, I get confused again.

  [1]: http://kilby.stanford.edu/~rvg/154/handouts/decidability.html
  [2]: http://www.amazon.com/Introduction-Theory-Computation-Michael-Sipser/dp/0534950973