you can view partners as any kind of business contact, I.e. a supplier or a customer.

 User is someone that interacts directly with Odoo. A user can be assigned a partner record (I.e. your Customer can have a User to interact with your database and that user is linked to the partner).

Regarding internal/public/portal user:

An internal user will have full access to the backend system. When you have Odoo Enterprise this type of user will require a license.

A portal user can access the customer portal to see their Sales Orders etc

the public user is someone who visits your Odoo website without authenticating

 regards