After MySQL install via Brew, I get the error - The server quit without updating PID file

I found that it was a permissions issue with the mysql folder.

chmod -R 777 /usr/local/var/mysql/ 

solved it for me.

EDIT 2012/09/18: As pointed out by Kane, make sure the mysql database is properly set up before doing anything else. See “PID error on mysql.server start?” for more info.

Original answer kept for history's sake: It most likely is a permissions issue. Check /usr/local/var/mysql/*.err. Mine said:

120314 16:30:14  InnoDB: Operating system error number 13 in a file operation.
InnoDB: The error means mysqld does not have the access rights to
InnoDB: the directory.
InnoDB: File name ./ibdata1
InnoDB: File operation call: 'open'.
InnoDB: Cannot continue operation.
120314 16:30:14 mysqld_safe mysqld from pid file /usr/local/var/mysql/janmoesen.local.pid ended

I also had to do this:

sudo chown _mysql /usr/local/var/mysql/*

I ended up with completely reinstalling of mysql, and it finally worked out.

WARNING This will remove all of your databases, so make sure to save dumps first.

brew remove mysql
brew cleanup
launchctl unload -w ~/Library/LaunchAgents/homebrew.mxcl.mysql.plist
rm ~/Library/LaunchAgents/homebrew.mxcl.mysql.plist
sudo rm -rf /usr/local/var/mysql

brew install mysql
mysqld --initialize --explicit_defaults_for_timestamp
mysql.server start # no sudo!