We analyze dispersion measure (DM) variations of 37 millisecond pulsars in the nine-year North American Nanohertz Observatory for Gravitational Waves (NANOGrav) data release and constrain the sources of these variations. DM variations can result from a changing distance between Earth and the pulsar, inhomogeneities in the interstellar medium, and solar effects. Variations are significant for nearly all pulsars, with characteristic timescales comparable to or even shorter than the average spacing between observations. Five pulsars have periodic annual variations, 14 pulsars have monotonically increasing or decreasing trends, and 14 pulsars show both effects. Of the four pulsars with linear trends that have line-of-sight velocity measurements, three are consistent with a changing distance and require an overdensity of free electrons local to the pulsar. Several pulsars show correlations between DM excesses and lines of sight that pass close to the Sun. Mapping of the DM variations as a function of the pulsar trajectory can identify localized interstellar medium features and, in one case, an upper limit to the size of the dispersing region of 4 au. Four pulsars show roughly Kolmogorov structure functions (SFs), and another four show SFs less steep than Kolmogorov. One pulsar has too large an uncertainty to allow comparisons. We discuss explanations for apparent departures from a Kolmogorov-like spectrum, and we show that the presence of other trends and localized features or gradients in the interstellar medium is the most likely cause.