Modules are used for 3 different things in ruby. First is namespacing. Having class or constant definitions inside a module won't collide with classes or constants outside that module. Something like this
class Product
  def foo
    puts 'first'
  end
end
module Affiliate
  class Product
    puts 'second'
  end
end
p = Product.new
p.foo # => 'first'
p = Affiliate::Product.new
p.foo # => 'second'
The second use for modules is as a place to stick methods that don't really have a place anywhere else. You can do this inside a class too, but using a module sort of tells people reading the code that it is not meant to be instanciated. Something like this
module Foo
  def self.bar
    puts 'hi'
  end
end
Foo.bar #=> 'hi'
Finally (and the most confusing) is that modules can be included into other classes. Using them this way is also referred to as a mixin, because you are "mixing in" all the methods into whatever you are including.
module Foo
  def bar
    puts 'hi'
  end
end
class Baz
  include Foo
end
b = Baz.new
b.bar #=> 'hi'
Mixins are actually a way more complected topic then I am covering here, but going deeper would probably be confusing.
Now, to me, S3 seems to be something that really belongs in the controller, since controllers are usually the things dealing with incoming and outgoing connections. If that is the case, I would just have a protected method on application controller, since that will be accessible to all other controllers, but still be private.
If you do have a good reason for it being in the model too, I would go for a mixin. Something like  
module AwsUtils
private
  def S3
    AWS::S3::Base.establish_connection!(
      :access_key_id     => 'Not telling',
      :secret_access_key => 'Really not telling'
    )
    data = yield
    AWS::S3::Base.disconnect
    data
  end
end
If you put that in lib/aws_utils.rb, you should be able to use it by adding include AwsUtils in both your controller and your model. Rails knows to look for classes and modules in lib, but only if the name matches (in wide case). I called it AwsUtils because I know what rails will look for when it sees that (aws_utils.rb), and to be honest, I have no idea what it will need for S3Utils ;-)
Feel free to ask for more info if I wasn't clear on something. Modules tend to be one of those things in ruby that while amazing, are downright baffling to newcomers.