A benefit of using a database system like MySQL is that databases can be accessed over the internet (although you should note that a fast internet connection is required to make such a feat enjoyable: the wav files that are transferred into and out of the database are typically rather large files). The copy of Luscinia on your local computer keeps a list of locations where it searches for Luscinia databases. This list can be edited in the "Administration Panel".
The addresses are standard IP addresses. If you are using Luscinia locally (i.e., MySQL, your databases, and Luscinia are all on the same machine), you can enter "localhost" instead here.