The database can be on either the local machine (where you'll be running the application/tools), or on another machine. When you connect to the DB, you specify the hostname of the machine where the database lives; if you don't give a hostname, the local machine is assumed. From the docs (for DBD::mysql):

    The hostname, if not specified or specified as '',
    will default to an mysql or mSQL daemon running
    on the local machine on the default port for the
    UNIX socket.
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You'll need to look at the DBD::mysql docs, because there is no standard on the manner in which the hostname should be passed to the driver. The docs suggest this DSN:

    $dsn = "DBI:$driver:database=$database;host=$hostname;port=$port";
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