Sometimes when I run my code, a core dump file is generated when I terminate the program by Ctrl+\\. The file name is of the form core.*.
You can avoid creating a core dump file by writing code that doesn't crash :)
Seriously, core dumps are useful because you can see the state of the program when it crashed, for "post mortem" debugging. You can open them in gdb and inspect the state of your program (especially if it was built with debugging).
Core dumps usually get made if the program has a SIGSEGV (usually caused by invalid pointer dereferencing), SIGABRT (which would happen if you called abort(), or in C++ by the default terminate() handler for exceptions in destructors etc) or some other fault. You can also trigger them explicitly with the debugger or programmatically.
If you've fixed all the bugs and it's perfect, you can delete them. Also, if you've changed your program in any way (and recompiled it) then they will become useless as the debug info now won't match what's in the core dump, so you can delete them then too.