For the peakedness comparison between two Gaussian random fields about their mean functions, a necessary and sufficient condition is derived in this paper in terms of their covariance functions. interestingly, such a condition is also necessary and sufficient for the convex ordering between the two Gaussian random fields having identical mean functions. The relation to the equivalence of two Gaussian random fields is illustrated through some parametric examples. Necessary and/or sufficient conditions are given for the peakedness comparison and convex ordering between two elliptically contoured random fields. These conditions are applied to examine how certain parameters affect the peakedness of some Gaussian or elliptically contoured random fields.