I had reason to go looking for this today, having not done it in a long, long time (thanks to the delightful cPanel interface).
To change a password on behalf of a user, first sign on or “su” to the “root” account. Then type, “passwd user” (where user is the username for the password you are changing). The system will prompt you to enter a password. Passwords do not echo to the screen when you enter them.
You can also change your own password, by typing “passwd” (without specifying a username). You will be prompted to enter your old password for verification, and then a new password.