TWO_TASK (or LOCAL on Windows) is an environment variable used to specifies the default remote Oracle service. Connections will be made to a remote database (via SQL*Net) without specifying a service name (tnsnames.ora entry). It will appear to programs that they connect to a local database, while in fact they are routed to a remote database.
$ export TWO_TASK=oraservice1
$ setenv TWO_TASK oraservice1
Where 'oraservice1' is defined in the local TNSNAMES.ORA file or in an Names/LDAP server.
One would normally connect to a remote database by specifying a service name (connect string). for example:
$ sqlplus scott/tiger@connect_str
By setting TWO_TASK you can omit the connect string (it is implied):
$ TWO_TASK=connect_str; export TWO_TASK $ sqlplus scott/tiger