For many times, I find people working on schemes over DVRs, and prove theorems on such setting. For example, my latest experience is the "semi-stable reduction theorem" by Kempf, Knudsen, Mumford and Saint-Donat: they proved semi-stable reduction theorem first over $\mathbb{C}$, then turned to the situation over DVR. 


It is "natural" for me to work over $\mathbb{C}$, or over character $p$, or even over arbitrary schemes. But why people like to choose DVRs in particular? This question would break to two parts: 


(1) Is there any technical advantage to work over DVR?


(2) What is the importance to consider this special case? Does this come from the interests of number theory? I have no clue of that.



Why considering schemes over discrete valuation rings?
Well, varieties and schemes typically come in families. If you accept that, then it's natural to consider the simplest case where the base is one dimensional and regular. This includes geometers smooth curves, and things like $Spec\mathbb{Z}$ or $Spec \mathbb{Z}\_p$ of interest to number theorists. If you want to concentrate your attention to near a specific fibre, then you may as well localize to work over a DVR, which is a bit like working over a small disk... Not sure what else to say.


