The discovery of ferromagnetism in two-dimensional (2D) van der Waals (vdW) crystals has generated widespread interest. Making further progress in this area requires quantitative knowledge of the magnetic properties of vdW magnets at the nanoscale. We used scanning single-spin magnetometry based on diamond nitrogen-vacancy centers to image the magnetization, localized defects, and magnetic domains of atomically thin crystals of the vdW magnet chromium(III) iodide (CrI3). We determined the magnetization of CrI3 monolayers to be ≈16 Bohr magnetons per square nanometer, with comparable values in samples with odd numbers of layers; however, the magnetization vanishes when the number of layers is even. We also found that structural modifications can induce switching between ferromagnetic and antiferromagnetic interlayer ordering. These results demonstrate the benefit of using single-spin scanning magnetometry to study the magnetism of 2D vdW magnets.