Very very basically, "processes" are running programs. They're referenced by a number, usually called a "process ID" or "PID". That's the "1276" in the error message.
A "child process" is a process started by some other process (which is the "parent process"). There's really not much more to it than that, in this case.
What this error message means, is that the parent process tried to tell one of its children to end (probably by sending a SIGTERM signal to it) and that didn't work. So the parent process got mean and sent a SIGKILL.
(Some signals can be ignored. SIGTERM is technically one of those; it's a sort of "please exit now" message; a process should clean up and exit upon seeing it. SIGKILL is more like "DIE, foul process! DIEEE!!!", and can't be ignored or even handled, so the receiving process can't clean up or anything -- it just dies. Since it's so potentially messy, it's generally only used for cases where stubborn processes refuse to end politely.)