A Distribution List is a simple list containing zero or more email addresses (local or remote).
A Mailing List has a lot more features than a distribution list. For instance, a mailing list supports user subscription and removal from the list, modifying message headers to support discussion lists, banning people from lists, and so on.
Another advantage of Mailing Lists over Distribution Lists is that VPOP3 expands Mailing Lists in the background while it expands Distribution Lists immediately. This means that if you send a message to a Distribution List with a large number of local users, it may take a while for VPOP3 to respond to your email client, but sending to an equivalent Mailing List will have an almost instantaneous response.
Forwardings are not really 'lists', because they can only contain a single email address, but they are managed here because they are, essentially, just a special case of a Distribution List, but with only a single member.
ODBC Mailing Lists are only available in VPOP3 Enterprise.
An ODBC Mailing List is a list where all the members are retrieved from an external ODBC database. VPOP3 cannot manage the list members - that has to be done outside of VPOP3 - but the other features of Mailing Lists are supported.
A Group is a separate type of list which can be used for administrative purposes. For instance, you can set that all members of a particular group cannot send outgoing messages. You can also send emails to all of a group's members, as with a distribution list, but this will only work internally.
Some permissions (eg with calendar sharing in VPOP3 Enterprise) can use a group as the 'target' of the permission - for instance, you could set it so that all members of the 'sales' group have permission to read someone's calendar.
Each user can only be in at most one group, and this is set in the user's settings, rather than in the group settings.