What do helper and helper_method do?

helper_method is straightforward: it makes some or all of the controller's methods available to the view.

What is helper? Is it the other way around, i.e., it imports helper methods into a file or a module? (Maybe the name helper and helper_method are alike. They may rather instead be share_methods_with_view and import_methods_from_view)

reference

The method helper_method is to explicitly share some methods defined in the controller to make them available for the view. This is used for any method that you need to access from both controllers and helpers/views (standard helper methods are not available in controllers). e.g. common use case:

#application_controller.rb
def current_user
  @current_user ||= User.find_by_id!(session[:user_id])
end
helper_method :current_user

the helper method on the other hand, is for importing an entire helper to the views provided by the controller (and it's inherited controllers). What this means is doing

# application_controller.rb
helper :all

For Rails > 3.1

# application.rb
config.action_controller.include_all_helpers = true
# This is the default anyway, but worth knowing how to turn it off

makes all helper modules available to all views (at least for all controllers inheriting from application_controller.

# home_controller.rb
helper UserHelper

makes the UserHelper methods available to views for actions of the home controller. This is equivalent to doing:

# HomeHelper
include UserHelper