Why won't my mac mini connect to my email?

Today I tried to send an email and my mac said that it had lost my password I guess. I went into system preferences to re-enter my password and it said that it wasn't correct. I went into my email providers site and changed my password, then went back and entered the new password into my system preferences.

It still told me that my password was incorrect. I then tried to add my alternate email account and password, and it would not work. So, I know that it is not a problem with my email provider. How do I fix this? I have tried restarting my mac and banging my head against the wall. :) Nothing helps.