Sometimes, Apple Mail starts prompting you for the Gmail password. Sent mail goes to your Outbox and stays there. For some reason Apple Mail loses its send mail configuration. Here's how to fix that.
Click the Drop Down arrow next to "Outgoing Mail Server (SMTP):"
If you have more than one email account in the top list box, make sure you select the correct email account. Click on "Advanced" and re-enter your Google password. Sometimes that field is blank or the password gets corrupted. Either way entering the password again in this field usually fixes the problem.
First you need to open the Apple Mail client preferences. Go to the "Mail" menu item next to the Apple logo at the top left of the top status bar.
Select "Edit SMTP Server List" at the bottom of the drop down window in the "Account Information" tab on the "Accounts" window.