1. For administrators or privileged users
      mysql -u root mysql  
      mysql> UPDATE user SET password = PASSWORD ("new password") WHERE user = 'name' ;  
　　mysql> FLUSH PRIVILEGES;  
　　mysql> QUIT  

2. -u root mysql mysql  
　　mysql> SET PASSWORD FOR name = PASSWORD ('new password');  
　　mysql> QUIT  

(Phpmyadmin or  MYSQL-front etc. tools is alternative choise to manage user permissions, more intuitive and easy)

3. Optimal:
mysqladmin -u root "old password" "new password"