To kill, stop, or continue processes, Linux uses a special form of communication called "signals". By sending a certain signal to a process, you can get it to terminate, stop, or do other things. This is what you're actually doing when you type Control-C, Control-Z, or use the bg or fg built-ins -- you're using bash to send a particular signal to the process. These signals can also be sent using the kill command and specifying the pid (process id) on the command line:

$ kill -s SIGSTOP 16224
$ jobs -l
[1]- 16217 Running                 xeyes -center red &
[2]+ 16224 Stopped (signal)        xeyes -center blue

As you can see, kill doesn't necessarily "kill" a process, although it can. Using the "-s" option, kill can send any signal to a process. Linux kills, stops, or continues processes when they are sent the SIGINT, SIGSTOP, or SIGCONT signals, respectively. There are also other signals that you can send to a process; some of these signals may be interpreted in an application-dependent way. You can learn what signals a particular process recognizes by looking at its man page and searching for a SIGNALS section.