I try to connect to a remote Oracle database using the Windows tool SQLPLUS.exe. I get back the error "ORA-03113: end-of-file on communication," even though ping works and TNS pings works also.
The ORA-3113 error is a generic error indicating that your communications with the database were terminated for some reason. Unforunately, no further information is given. The first thing I'd do is to verify that the SQL*Plus client is Oracle 8i or higher and that the database is Oracle 8i or higher. If one of those is lower than Oracle 8i, then you can have difficulties.
After that, I'd contact your database administrator. He may be able to locate a trace file from your session which provides further clues. If the DBA is unable to track down the problem, then you will need to contact Oracle Support for further assistance.