contacts.js

jQuery(function($) {
  $('#search').keyup(_.debounce(function(e) {
    e.preventDefault();
    $.get($(this).data('search_contacts_path'), {
      search : $(this).val(),
    });
    }, 200)
  );
});

// debounce is from _underscore.js which delays the function call. This is optional.

contacts_controller.rb

class ContactsController
  def search
    respond_to do |format|
        format.json {
          @contacts = Contacts.search(params[:search]) # whatever your search stuff is
          render :json => @contacts.to_json
        }
    end
  end
end

index.html.erb

<%= text_field_tag :search, params[:search],
  :placeholder => "Search...",
  'data-search_contacts_path' => search_contacts_path %>