A real algebraic variety is a set of points in real Euclidean space that satisfy a system of polynomial equations.  Metric algebraic geometry is the study of properties of real algebraic varieties that depend on a metric. In this talk, we introduce metric algebraic geometry through a discussion of Voronoi cells, bottlenecks, and the reach of an algebraic variety. We describe applications to the study of the geometry of data with nonlinear models and to low-rank matrix approximation.