In derived algebraic geometry there are several different setting,i.e., sometimes we use $E_{\infty}$ring,somethings we use dgalgebra,... It is for different situations. But could someone give some examples illustrating under what problem we use relevant "derived structure" ($E_{\infty}$ ring, dgalgebra...)? Some motivations?

Here are the two motivations I know of: Number 1 comes from algebraic topology. The definitive reference is http://www.math.harvard.edu/~lurie/papers/survey.pdf , which explains it much better than I can. Very roughly, complex oriented cohomology theories are represented by Eoo rings and are classified by their grouplaws. The group law is what the cohomoly theory does to the tensor product of line bundles. Some interesting group laws come from elliptic curves. So if you are very daring, you can take the moduli space of elliptic curves, and assign to each point the Eoo ring representing the cohomology theory fitting to the group law of the elliptic curve. The result should be a sheaf of Eoo rings on the moduli space of elliptic curves, or in other words a derived structure of Eoo rings. Number 2 comes from virtual fundamental classes in algebraic geometry. Kontsevich in section 1.4 of http://arxiv.org/pdf/hepth/9405035 suggested that for specific types of moduli spaces (those with a perfect obstruction theory) their should exist a derived structure of dgalgebras. The dgalgebra structure should come from local presentations as intersection of submanifolds as discussed in this question: Serre intersection formula and derived algebraic geometry? . Kontsevich suggested that via a RiemannRoch formula you should get the virtual fundamental class needed in DonaldsonThomas theory and GromovWitten Theory. 

