A computational system is Turing complete if certain features of its operation can reproduce those of a Turing machine, which is a sort of bare-bones abstracted model of the low-level process of computation. This is important because you can, in principle, simulate the active parts of any Turing complete system in any other Turing complete system (though doing so will be inefficient in a lot of cases); in other words, if you've got enough time and memory, you can calculate anything calculable with any system meeting a fairly minimal set of requirements. Thanks to this result, we know that there's a deep symmetry between different flavors of computation that might not otherwise be obvious. There are some caveats, though: in particular, the idealized version of a Turing machine assumes infinite memory.

Now, to answer your actual question, the branch of mathematics that this comes from is called computability theory, and it's related to the study of mathematical logic and formal languages. The textbook I got most of my understanding of it from is Hopcroft, Motwani, and Ullman's Introduction to Automata Theory, Languages, and Computation, although it might be worth looking through the "Best Textbooks on Every Subject" thread to see if there's a consensus on another.