Let \(\Delta\) be the partition of a region in affine space into a finite number of polyhedral cells. A \textit{multivariate spline} with smoothness \(\mu\) over \(\Delta\) is a \(C^\mu\)-function whose restriction to each cell of \(\Delta\) is a polynomial. A \(C^\mu\) \textit{piecewise algebraic variety} is the zero set of a collection of multivariate splines with smoothness \(\mu\). The article serves primarily as a review of the algebraic structure of multivariate spline spaces and of the concepts from algebraic geometry that translate to the more general setting of multivariate splines, namely the correspondence between ideals and varieties. In addition, although the Hilbert Nullstellensatz does not hold in general for splines, the authors provide some instances in which it does. The article targets a general mathematical audience familiar with the basic concepts of algebraic geometry.