Possible security problem in Ubuntu 9.04?

[Dick Stuart-Grenville]

Being new to Ubuntu, and very likely the cause of the problem, I see 
at top right of the desktop my full name (not my log-in name) followed 
immediately by my supposedly secret password.
It looks wrong - and certainly insecure - so I wish to remove said 
password from the desktop. How do I do this safely, and without 
causing any log-in difficulty?