TL;DR:
Run this as root   and you'll be all set:
rm $(grep socket /etc/my.cnf | cut -d= -f2)  && service mysqld start
Longer version:
You can find the location of MySQL's socket file by manually poking around in /etc/my.conf, or just by using
grep socket /etc/my.cnf | cut -d= -f2
It is likely to be /var/lib/mysql/mysql.sock.  Then (as root, of course, or with sudo prepended) remove that file:
rm /var/lib/mysql/mysql.sock
Then start the MySQL daemon:
service mysqld start
Removing mysqld will not address the problem at all.  The problem is that CentOS & RedHat do not clean up the sock file after a crash, so you have to do it yourself.  Avoiding powering off your system is (of course) also advised, but sometimes you can't avoid it, so this procedure will solve the problem.