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?

$\begingroup$ In mirror symmetry one of the things we are interested in is deformations of the relevant dg/Ainfty algebras/categories. The relevant deformation spaces are, or should be, related to, for instance, quantum cohomology (see BarannikovKontsevich or Kontsevich's original homological mirror symmetry paper). As for Einfty rings, see Lurie's survey on elliptic cohomology for some motivation. $\endgroup$ – Kevin H. Lin Jul 26 '10 at 8:53

$\begingroup$ At least in the context of ToenVezzosi's HAG (vol 2, if I remember correctly), derived algebraic geometry is specifically simpliciallyenriched algebraic geometry, while they use "complicial algebraic geometry" and "brave new algebraic geometry" for enrichment in complicial sets and spectra respectively. $\endgroup$ – Harry Gindi Jul 26 '10 at 8:58

5$\begingroup$ A frequent remark is that dgas are only good in characteristic zero, while in positive characteristic one has to use simplicial rings or E_k or E_\infty rings. See math.harvard.edu/~lurie/papers/moduli.pdf for a case, where this thought is worked out. $\endgroup$ – Lennart Meier Jul 26 '10 at 12:46
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.

$\begingroup$ I minor point is that complex orientations are not necessarily represented by $E_{\infty}$ ring spectra. Having an $E_{\infty}$ complex orientation is something one has to work for. See recent work of Niles Johnson and Justin Noel or the thesis of Matt Ando or Jim McClure. $\endgroup$ – Sean Tilson Oct 17 '12 at 21:56