After a process receives a SIGSTOP it is typically described as "suspended", since the scheduler is no longer allocating CPU cycles to the process but otherwise it is business as usual.
SIGSTOP holds a special place in the Unix signal pantheon, as it is one of the only two signals that a process is NOT allowed to catch, block, or ignore. The only other signal that has this special status is SIGKILL.
One means of sending the SIGSTOP signal to a process is through your shell. All commonly used Unix shells follow the convention of interpreting the Control-Z (i.e. ^Z) key stroke as an instruction to send SIGSTOP to the current foreground process, causing it to suspend and allow the shell to accept new commands. The process can then be reactivated (by sending a SIGCONT) using the shell builtin fg (which returns the process to the foreground) or bg which causes the reactivated process to become a background process.