I am having trouble discerning the difference between a double integral and an iterated integral. I have seen alternate notation to the effect of: 
\begin{align\*}
\iint\limits\_R f(x,y) \ \ dR = \int\_a^b \int\_c^d f(x,y) \ \ dx \ dy = \int\_a^b \left(\int\_y^{y^2} f(x,y) \ dx\right) \ dy
\end{align\*}
My question is: what is the difference between these two types of integrals, and do the above integrals I've written represent one or both of these? 

Thanks.