A core file will be from a core dump, which is produced when a program crashes. In one of your systems setup files, such as /etc/profile, you may find a line that looks like:
ulimit -c 20
(type in "help ulimit" in the bash shell for a manpage or info bash for more information). This value may be read in from a config file in /etc/sysconfig and then the value used in a ulimit command.
The number is the limit of core dumps in 1024 byte blocks. You can also filter out core files from being backed up. They are only useful when you use a debugger. If you don't plan on doing that, you can even delete it. The timestamp on the core dump might be useful. It will tell you when the crash occurred. Being on the root directory,you know that a root program crashed.