root: is a special account which is usually locked in Ubuntu. This account is in effect the ultimate super user and can change anything on the system. If you want to enable this account you can but its not recommended see How to enable root login? For how this can be done and a discussion on why its not recommended.

user: These are the users of your systems there are three types administrators, normal users and guest. Administrators can make major changes to the system while normal users can not. There is a useful summary in the RootSudo Comunity Documentation. The guest account is used to provide casual access to someone (to play a game, surf the web, etc.) without giving them access to other users files. Any files they create are automatically removed when they log out.

groups A user can be a member of one or more groups groups are used to control privileges within the system for example all administrators are in the sudo group but you can also create you own groups. You may for example have a sales team who need to share some files but you want to prevent other people from accessing these files. By creating a special group for these people you can do that. See:How can I share a directory with an another user? for how you can set this up.