'Can't connect to local MySQL server through socket'

[just2blue4u]

002053:20060609:084936 Starting zabbix_server. ZABBIX 1.1.
002053:20060609:084936 Failed to connect to database: Error: Can't connect to local MySQL server through socket '/var/lib/mysql/mysql.sock' (111)
002635:20060609:085004 File [/var/run/zabbix/zabbix_server.pid] exists. Is this process already running ?

This was standing in the zabbix log files. The Zabbix server has been down since then. When i tried to restart, the error

File [/var/run/zabbix/zabbix_server.pid] exists. Is this process already running ?

came up again.
After removing the file manually, i was able to start zabbix again.

- Why was the server down?
- Why isn't the pid-File removed by zabbix?