Wolfram Mathematica is a desktop environment for building applications in the Wolfram Language. The Wolfram Language, in turn, is a functional programming utility for solving all sorts of complex problems in a wide variety of fields, from science to business and government.
Ours is a consultancy that uses tools like Mathematica to solve insanely hard problems for our clients, which are mostly large corporations with complex operations. Everyone at our small firm uses Mathematica in one way or another.
While Mathematica can quite literally do anything, the most common use case is as a means to build a mathematical model or representation of some complex system that one wishes to study using data. Along the way the model builder can use the vast array of internal functions (6500 or so at last count) to perform operations as diverse as image processing, statistics, and optimization. The real benefit of Mathematica is in its ability to allow model builders to generate such models very quickly with less code than other languages like Python or R.