The Product Category (or Internal Category as it is called in V8) is used to define the default expense and income accounts (and stock accounts for valuation). You can define a product category according to the way in which you want to see your turnover in Odoo, e.g. to replace separate income / expense accounts. Usually, you create the product categories from an accounting / reporting point of view.

The Public Product Categories are used to group products on your website. These categories are not the same as the product categories, because they tend to be created from a different point of view. The way in which you display your reporting, does not necessarily comply with the way you want to classify products for your customers.

Example: you define a Product Category Printers. On your website, however, you would like your customers to be able to search for laser printers, inkjet printers, etc. Therefore, you define Public Product Categories to make this classification work.

The Pos Product Categories are used to classify your products in your point of sale. The categories you define, will be displayed at the top of your POS screen.

Product Categories are mandatory in Odoo. For the other two categories, you can choose to use or to not use them. They have to be created separately, but you can import them from a csv file. The structure can be as simple or as complex as you like or need.