I've found thes errors in the mysql err log for my database and I don't know how to run the functions requested.

Does this mean the something (me?) changed the "format" for passwords, or does it just mean the password itself was changed?

060108 6:49:41 [Warning] mysql.user table is not updated to new password format; Disabling new password usage until mysql_fix_privilege_tables is run
060108 6:49:41 [Warning] Can't open and lock time zone table: Table 'mysql.time_zone_leap_second' doesn't exist trying to live without them

Can someone tell me where and how to run the "fix" function specified above and how to create the missing Table 'mysql.time_zone_leap_second'?

You can't, but we will... its a problem on our side. MySQL 5 had its MySQL User Database Structure updated but we are still using the old structure. We will be sure to change it on the next version