This is an example of how I used the jQuery Automcomplete Widget with Rails 3.2 for a recipe search field. Some of the jQuery stuff mentioned below is included out of the box with a new Rails application.

gem 'jquery-rails'
//= require jquery
//= require jquery_ujs
//= require jquery-ui
*= require jquery-ui-1.8.18.custom.css
def search_on_title
    # search_on_title is a method on the model to do a wildcard search on keyword
    recipes = Recipe.search_on_title(params[:term], params[:category_id])
    render json: recipes.map(&:title)
  end
resources :recipes do
    get 'search_on_title', :on => :collection
  end
<%= form_tag '', :method => :get do %>
    <%= search_field_tag :term, params[:term], :class=> "input-large" %>
    <%= submit_tag "Filter", :name => nil, :class => 'button btn', :id => "search_bn" %>
  <% end %>         
$("input#term").autocomplete({
    source: '<%=search_on_title_recipes_path%>?category_id=<%=params[:category_id]%>',
    minLength: 2, delay: 500
});

There’s also a Rails gem to make autocomplete easier, but I found this to be easy enough such that I didn’t have to depend on an additional plugin.