It's a bit hard to answer without knowing your background or interests... the literature is pretty vast. Nevertheless, here are a few jumping-in points:
"Moduli of curves" by Harris and Morrison carefully avoids using the word stack throughout, but they give a very nice (somewhat informal) discussion of stable reduction. And you'll learn a lot about moduli of curves, which is presumable what you are interested in. For a more formal and arithmetically minded treatment of stable reduction, take a look at the last chapter of Qing Liu's "Algebraic geometry and arithmetic curves".
The book "Fundamental algebraic geometry" by Fantechi et al. is not directly related to moduli of curves. But the section written by Vistoli will in particular tell you exactly what a stack is. There are also many more informal introductions to stacks floating around online, like one by Fantechi ("Stacks for everybody") and Heinloth ("Lectures on the moduli stack of vector bundles on a curve") ... you might find these useful, too.
You may also be interested in the second volume of "Geometry of algebraic curves" by Arbarello, Cornalba and Griffiths.
Hope this helps.