How to know a function is integrable or not? Let say 
$$ h(x) = \begin{cases}x^2,& x \in \mathbb {Q}\\-x^2,& x \notin \mathbb{Q}\end{cases}
           $$
Is there a difference between riemann integrable and integrable?

And can I just say contiuous function is integrable and hence the above function is not
integrable in [-1,1]?

The function is equal almost everywhere to $k(x) = -x^2$, and (as you note) continuous functions are integrable.  So your function is Lebesgue integrable.  
Together with Dimitrios's answer we see: there is a difference between Riemann integrable and Lebesgue integrable.
