There are possible solutions here: http://forums.mysql.com/read.php?35,64808,254785#msg-254785 and here: http://forums.mysql.com/read.php?35,23138,254786#msg-254786

All of these are config settings. In my case I have two computers with everything in XAMPP synced. On the other computer phpMyAdmin did start normally. So the problem in my case seemed to be with the specific computer, not the config files. Stopping firewall didn't help.

Finally, more or less by accident, I bumped into the file:

...path_to_XAMPP\XAMPP...\mysql\bin\mysqld-debug.exe

Doubleclicking that file miraculously gave me back PhpMyAdmin. Posted here in case anyone might be helped by this too.

I once had this problem and solved it by installing mysql-server, so make sure that you have installed the mysql-server, not the mysql-client or something else.

That error means the file /var/run/mysqld/mysqld.sock doesn't exists, if you didn't install mysql-server, then the file would not exist. So in that case, install it with

sudo apt-get install mysql-server

But if the mysql-server is already installed and is running, then you need to check the config files.

The config files are:

/etc/my.cnf
/etc/mysql/my.cnf
/var/lib/mysql/my.cnf

In /etc/my.cnf, the socket file config may be /tmp/mysql.sock and in /etc/mysql/my.cnf the socket file config may be /var/run/mysqld/mysqld.sock. So, remove or rename /etc/mysql/my.cnf, let mysql use /etc/my.cnf, then the problem may solved.