If you have changed your user password that you use email services on you'll need to update your mail clients too. To update your mailbox password in Apple's Mail client, open Mail.
If you have recently changed your password Mail should notice that its attempt to login to your mailbox has failed and you should be prompted for the new password. Simply enter it and select Remember password.
If this doesn't happen for any reason, click Mail and then Preferences.
Click the Accounts tab. Select your mailbox from the accounts list. Under Account Information select the password field and type in your new password.