I know that compact space is countably compact, and every compact is pseudo-compact.
Is there a simple example that show any Compact space is not necessarily compact ?
Is there an example to show that any pseudo-compact space is not necessarily compact?
Okay, so, your initial statement is incorrect. The correct implications are as follows: $$\text{compact}\;\;\implies\;\;\text{countably compact}\;\;\implies\;\;\text{pseudocompact}$$ (Note that "countably compact" is not the same as "$\sigma$-compact.") To see that the reverse implications do not hold in general, consider the first uncountable ordinal endowed with the order topology which is countably compact space but not compact.
An example of a pseudocompact space which is not countably compact is harder to construct, but a good one can be found here.