Those are two different (but similar) scripts that allow you to connect to MySQL databases, browse, view, export, import, made changes, updates, delete etc. the content of that MySQL database.
PHPMyAdmin is most known script of that kind and got lots of features (can be downloaded at sorceforge for example - just search in Google).
110DBManager is a custom 110MB script made by 110MB admins and programmers (download:
http://www.110mb.com/phpmyadmin/ ) based on phpMiniAdmin (other, very simple script for the same thing).
There are also other scripts.
You don't need them if you have no (and/or don't use) a MySQL database. But if you buy a MySQL upgrade here on 110mb you can choose which software to use.
edit:
general vegitable was faster.

But I have to correct one thing.
(...) phpmyadmin is not allowed on 110mb as it is resourse intensive, (...)
The policy has changed. PHPMyAdmin is allowed at the moment to use on accounts that have paid MySQL upgrade.