Okay I think I have it working now. I guess the problem was the uppercase vs lowercase BORG vs borg stuff in the tnsnames and listener.ora files. Here is what I have in the end:

listener.ora
Code:
LISTENER =
  (DESCRIPTION_LIST =
    (DESCRIPTION =
      (ADDRESS = (PROTOCOL = TCP)(HOST = borg.ourdomain.com)(PORT = 1521))
    )
  )
tnsnames.ora
Code:
borg =
  (DESCRIPTION =
    (ADDRESS = (PROTOCOL = TCP)(HOST = borg.ourdomain.com)(PORT = 1521))
    (CONNECT_DATA =
      (SERVER = DEDICATED)
      (SERVICE_NAME = borg)
    )
  )
And then, as a normal user (not root or oracle user) I run the following command:

Code:
> sqlplus system
Then input my password for the user 'system' and then it successfully gives me the SQL prompt.

However, if I try this command:

Code:
> sqlplus system@borg
or
Code:
> sqlplus system@borg.ourdomain.com
It asks me for the password and then gives the following error again:

Code:
ERROR:
ORA-12504: TNS:listener was not given the SERVICE_NAME in CONNECT_DATA
Got any clues why?