Why, from a string theory perspective, is it natural to consider the Deligne-Mumford (resp. Kontsevich) compactification of the moduli of curves (resp. maps [from curves to a target space X]) rather than some other compactification? 


In any case, what other compactifications of the moduli of curves have been studied? Similarly, what other compactifications of the moduli of maps have been studied? Do any of these other compactifications lead to an interesting "Gromov-Witten theory"?



Gromov-Witten theory and compactifications of the moduli of curves
I can give individual answers to a lot of your questions, but I can't answer any of them completely, nor can I fit all these answers together into a coherent whole.


For string theory, there does seem to be something special about the Deligne-Mumford compactification. Morally, what's going on is this: string theorists are allowing cylinder-shaped submanifolds of their Riemann surfaces to become infinitely long. The only finite energy fields on such infinitely long submanifolds are constant, so you can replace the long cylinder with a node. (Likewise, morally, if you allow vertex operators at two marked points to come together, you should take their operator product. This is what bubbling when marked points collide does for you.)


Somewhat more technically: The first step in (bosonic) string theory is to compute the partition function of the nonlinear sigma model as a function on the space of metrics on your worldsheet. This function on metrics descends to a section of some line bundle on the moduli stack of complex structures on the worldsheet. When you can compute it at all, you can show that this section has exactly the right pole structure it needs to be a section of the 13th power of the canonical bundle tensored with the 2nd power of the dual of the line bundle corresponding to the boundary divisor of the Deligne-Mumford compactification. (There's an old Physics Report by Phil Nelson that explains this pretty well, although not with anything you'd call a proof. Should also credit Belavin & Knizhik, who did the initial calculations.)


There's a somewhat more modern perspective on this (Zwiebach, Sullivan, Costello,...) that says that the generating function of string theory correlation functions for smooth Riemann surfaces satisfies a certain equation (a "quantum master equation"), which gives instructions for how to extend the theory to nodal Riemann surfaces. Different master equations give different recipes for extending to the boundary, if I understand your advisor correctly.


People have played around with other compactifications. There are a lot of different compactifications of the stack of smooth marked curves. People have already mentioned a few of them. David Smyth has some cool results which classify the "stable modular" compactifications of the stack of curves (<http://arxiv.org/abs/0902.3690> ). For compactifications of the moduli of maps, the only one that comes immediatley to mind is Losev, Nekrasov, and Shatashvili's "freckled instanton compactification", in which IIRC, you allow zeros and poles to collide and cancel each other out. 


