A god account can usually do anything system-wise, from adding more users to changing anybody's password to complete system reconfiguration. As a hacker, this is typically your objective. Special accounts are usually either accounts used by the system itself or accounts that fulfill some type of administrative roll without full god access. A good example of this is anonymous ftp. Typically guest accounts have fairly restrictive access to the system, especially on publicly accessible systems.
Groups are simply groupings of users.
They are primarily used to ease system administration. For example, instead of having to assign access to a new hard drive to the forty accounting users, an admin just has to assign the accounting group the access. Even special privileges can often be assigned by group, such as the ability to manage a set of programs or system functions like printing. Most modern systems allow accounts to belong to more than one group.