Syntax:
PASSWORD(str)

Returns a hashed password string calculated from the cleartext password
str. The return value is a nonbinary string in the connection character
set (a binary string before MySQL 5.5.3), or NULL if the argument is
NULL. This function is the SQL interface to the algorithm used by the
server to encrypt MySQL passwords for storage in the mysql.user grant
table.

The old_passwords system variable controls the password hashing method
used by the PASSWORD() function. It also influences password hashing
performed by CREATE USER and GRANT statements that specify a password
using an IDENTIFIED BY clause.

URL: http://dev.mysql.com/doc/refman/5.5/en/encryption-functions.html

