What is $\mathcal{N}=2$ QED? I would like to know is $\mathcal{N}=2$ QED is simply a $\mathcal{N}=2$ theory with gauge group $U(1)$ like in normal QED? If not, exactly what theory is it? Is there some reference for it? 
 A: When people talk about $\mathcal{N}=2$ QED in 4d I think they normally mean a $U(1)$ gauge theory (one $\mathcal{N}=2$ vector multiplet) coupled to one or more hypermultiplets (usually all with the same $U(1)$ charge). As an example of this usage see Witten's discussion of $\mathcal{N}=4$ QED in 3d (which can be obtained by dimensional reduction from the $\mathcal{N}=4$ 4d theory) in this paper.
Similarly $\mathcal{N}=1$ QED consists of a ($\mathcal{N}=1$) vector multiplet coupled to chiral multiplets. To get from this to the $\mathcal{N}=2$ theory you need to pick an odd number of chiral multiplets. One chiral multiplet needs to be neutral under the $U(1)$ gauge symmetry (this will combine with the $\mathcal{N}=1$ to form the $\mathcal{N}=2$ vector multiplet), and the other ones should be paired up so the half of them have $U(1)$ charge, eg., $+1$, and the other ones have charge $-1$. Finally you need to add a cubic superpotential with the correct coefficient. See, for example, section 12.5 in the review by Sohnius.
