Click On a Link to Update Database Record. How Do You Do It?

Learn about controllers and actions in Rails. I would recommend the Agile Web Development with Rails or Ruby for Rails as good books to start with.

Eventually you'll learn about the link_to helper you can use in your views, for example:

<% link_to 'Complete This To-Do Item', :action => 'completed' %>

and you have to implement the corresponding 'completed' action in your controller. But that's a lot to explain here, so start with one of the books above.