Should you upgrade your Drupal website to version 8?

When should I move a Drupal 7 site to Drupal 8?

You should upgrade if you want a website that is responsive to mobile devices.  A responsive theme is no longer an option, a theme update can help your site stay relevant and useful to an increasingly mobile browsing community.

According to drupal.org: "There’s no site maintenance reason to upgrade right away. Drupal 7 will be supported long after Drupal 8 is officially released. But if there’s a new or improved feature in D8—like fully translatable admin interfaces, or native schema.org markup—that you think could make your work better, take advantage of it."

D8upgrade is a free Web service that makes it as easy as possible to generate an upgrade report of your site. Simply type your website's URL and an email address, and the service will send you the results. No special access or login required.

When should I move a Drupal 6 site to Drupal 8?

You should upgrade now. As of February 24, 2016, Drupal 6 has reached end-of-life and will no longer be supported.  Drupal’s security team only supports the two most recent major versions. As Drupal 6 is no longer maintained, people with D6 sites definitely have to upgrade, or look for an alternate long term support solution: security threats discovered in Drupal 6 will no longer be systematically addressed.