# ps | grep safe
14592 root 136m S /tmp/data/safe/safe
16210 root 1664 S grep safe
# ps | grep safe\$
14592 root 258m S /tmp/data/safe/safe
So what does \$ mean? Is it a regular expression?
Yes, it is a regex. $ is a regex character that means "end of line". So by saying grep safe\$ you are grepping all those lines whose name ends with safe and avoiding grep itself to be part of the output.
The thing here is that if you run the ps command and grep its output, the grep itself will be listed:
$ ps -ef | grep bash
me 30683 9114 0 10:25 pts/5 00:00:00 bash
me 30722 8859 0 10:33 pts/3 00:00:00 grep bash
So by saying grep safe\$, or its equivalent grep "safe$", you are adding a regular expression in the match that will make the grep itself to not show.
$ ps -ef | grep "bash$"
me 30683 9114 0 10:25 pts/5 00:00:00 bash
As a funny situation, if you use the -F option in grep, it will match exact string, so the only output you will get is the grep itself:
$ ps -ef | grep -F "bash$"
me 30722 8859 0 10:33 pts/3 00:00:00 grep -F bash$
The typical trick to do this is grep -v grep, but you can find others in More elegant "ps aux | grep -v grep". I like the one that says ps -ef | grep "[b]ash".