Personal settings include your Artifactory API Key, password and email address.
If the Can Update Profile field is enabled, you can update all of your profile details except for the password, since only an administrator can update the password.
Note that there may be cases in which you want to leave this option disabled to prevent users from updating their profile. For example, a departmental user with a single password shared between all department members.
You are not able to change your password if the system is configured to use external authentication such as LDAP. For example, if users from LDAP are not created internally (i.e. Administration | Security | LDAP | LDAP Settings and the Create Internal Users is disabled.