Make core files writeable?

[Rory Campbell-Lange <rory@campbell-lange.net>]

I have been asked by an application developer, who is trying to find out
problems with one of their products running on our server, to set up the
server so that "core files can be written". What are these, and how would I
do this?

I have also been instructed to, once a core file is generated, to "rename
the core file to a unique name and collect a stack backtrace".

1. How will I know when a core file has been generated?
2. What is and how do I do a stack backtrace?

Apologies for these basic questions; and thanks for any help.
Rory