From what I can see, cookies in Chrome have been disabled. To be able to login to Facebook without entering the password each time, cookies must be enabled and the, 'Remember password' box, must be checked. This way, the browser will be able to have your Facebook credential for logins saved on the login page. Below, are the instructions on how to enable cookies in Chrome.

Chrome

Steps are the same for both PC and MAC.
1.Click on the Chrome menu button on the browser bar.
2.Click "Settings."
3.Scroll down and click "Advanced."
4.In the Privacy and security section, click Content Settings.
5.Click Cookies.
6.Click the slider to "Allow sites to save and read cookie data (recommended)."