Categorical data are data that track a characteristic in which people are put into non-quantifiable, mutually-exclusive categories.

If I want to measure if you're from New York, you either are or you aren't. You can't be 0.7 New Yorks. Everyone exists in one of two categories, from New York and not from New York.

We can split categorical data into two subtypes, ordinal (order matters) and nominal (order doesn't matter). The "from New York/not from New York" is an example of nominal data. If I wanted to put people into categories based on their income (e.g., $0-$10000, $10001-$20000, etc.) it would be ordinal. The information being tracked is still not quantitative (we're just tracking your membership in a category), but there is an order to the categories.

The terms "polynomial" and "binomial" do not make sense to use in the context of categorical data. It sounds like you may just be using them to refer to nominal data that track 2 categories vs. more than 2 categories. The former is often referred to as a "binary variable" because it has two states (e.g., "In New York" and "Not in New York").