Ctrl + C doesn't always work to kill the current process (for instance, if that process is busy in certain network operations). In that case, you just see "^C" by your cursor and can't do much else.
What's the easiest way to force that process to die now without losing my terminal?
Summary of the answers:
Usually, you can Ctrl + Z to put the process to sleep, and then do
kill -9 _process-pid_, where you find the process's pid with ps and other tools.
On Bash (and possibly other shells), you can do
kill -9 %1 (or '%N' in general) which is easier. If Ctrl + Z doesn't work, you'll have to open another terminal and kill from there.