The “nom” in “nominal’ comes from the Latin for “name” (not “number”) like in the word “nomenclature.” So a nominal variable is a variable whose values are named. An examples might be “red,” “blue,” “green,” etc. if the dependent variable were color, or “American,” “British,” “French,” etc. if the dependent variable were nationality. The key idea here is that there are categories and that they do not have a natural numerical order. Compare with ordinal variables where there is a natural order; e.g., “bad,” “mediocre,” “good,” and “great.”
When there are just two categories, there is essentially no difference between nominal logistic regression (where where the dependent Y takes on only the values of 0 and 1), except that, because there is no natural ordering, the decision of which of the values of the nominal variable to assign to 0 and 1 is completely arbitrary.
When there are more than two categories for the nominal variable, then the multinomial distribution would model the chance of selection of a particular category and multinomial logistic regression should be used.
Note that there are a number of other terms that are used for “nominal” variables. “Attribute variables” and “categorical variables” are other commonly used terms. The idea of “classification” is also similar.