Metric algebraic geometry is a term proposed for the study of properties of real algebraic varieties that depend on a distance metric. The distance metric can be the Euclidean metric in the ambient space or a metric intrinsic to the variety. In this talk, we introduce metric algebraic geometry through a discussion of Voronoi cells, bottlenecks, offset hypersurfaces, and the reach of an algebraic variety. We also show applications to the computational study of the geometry of data with nonlinear models.