When SSL is enabled in Magento, the store automatically switches to a secure connection for sensitive parts of the site, including the Customer login page, and the checkout.

The first step is to install an SSL certificate on the hosting account. See Here for instructions on how to purchase and set up an SSL certificate.

Once this is done, verify that SSL is working by loading https://YOURSITE/LICENSE.html . You should see the Magento License, with the browser displaying its SSL identifier (e.g. Padlock etc). If you see an error, SSL is not set up correctly on the hosting account. 

When SSL is working correctly, Magento needs to be told to use secure URLs. To do this:

1. Log in to the admin backend
2. Go to System -> Configuration -> Web
3. Change http to https for the Secure Base URL, preserving the domain otherwise, including the trailing slash
4. Switch the "Use Secure URLs in Frontend" and "Use Secure URLs in Admin" drop down boxes to Yes.

Once this is done, confirm that SSL is working as it should by clicking the Log In link on the store. This is sometimes called My Account or similar. When working the URL will change from http to https, and the browser should indicate that you are viewing a secure page.