What's the difference between categories and piece medium/material?

Medium is more self explanatory - it's what your piece is made of! The dropdown menu you see on EntryThingy has 33 pre-selected options for the Piece Medium. When you go to add your art, you will see the option to select what the piece if made out of. It looks like this:

Categories are set by the gallery. They can be anything! It's dependent on the gallery. Some galleries use which type of entry you have (e.g. members vs. non-members), some do which juror category, and some do use the medium here (e.g. if they have different voting categories for painting vs. sculpture). You might have or see a show that only has one category - it just depends on the show.