During parameter estimation in Generalized
Linear/Nonlinear Models, you may occasionally encounter a warning
message that a "zero pivot element" was detected. This message
generally means that during parameter estimation a redundancy was detected
in the design matrix. In other words, one of the variables or columns
in the design matrix (predictors) is redundant with other columns (predictors).

This of course will usually happen when categorical predictors are coded using the overparameterized model (see GLM Introductory Overview - Sigma-Restricted and Overparameterized Model). For example, if you have a categorical predictor, "Gender," with two values, "Male" and "Female," in the overparameterized model, the program would create two indicator columns in the design matrix: One for males (1=yes, 0=no), and one for females (1=yes, 0=no). Obviously, these two columns are redundant and, hence, during estimation the program will drop one of them.

This mechanism is equivalent to the application of the generalized inverse in GLM to automatically remove redundant columns from the design matrix.

Note that in some cases this message may occur with continuous predictors as well, or when the overparameterized model is not being used. In this case, it simply means that some predictors are entirely redundant with other predictors and, thus, unique parameter estimates for those predictors cannot be computed (and they are zeroed instead).