build.sh

elm-make Spelling.elm --output=spelling.js

index.html

<div id="spelling"></div>
<script src="spelling.js"></script>
<script>
    var app = Elm.Spelling.fullscreen();

    app.ports.check.subscribe(function(word) {
        var suggestions = spellCheck(word);
        app.ports.suggestions.send(suggestions);
    });

    function spellCheck(word) {
        // have a real implementation!
        return [];
    }
</script>

Spelling.elm

port module Spelling exposing (..)

import Html exposing (..)
import Html.App as App
import Html.Attributes exposing (..)
import Html.Events exposing (..)
import String


main =
  App.program
    { init = init
    , view = view
    , update = update
    , subscriptions = subscriptions
    }


-- MODEL

type alias Model =
  { word : String
  , suggestions : List String
  }

init : (Model, Cmd Msg)
init =
  (Model "" [], Cmd.none)


-- UPDATE

type Msg
  = Change String
  | Check
  | Suggest (List String)


port check : String -> Cmd msg

update : Msg -> Model -> (Model, Cmd Msg)
update msg model =
  case msg of
    Change newWord ->
      ( Model newWord [], Cmd.none )

    Check ->
      ( model, check model.word )

    Suggest newSuggestions ->
      ( Model model.word newSuggestions, Cmd.none )


-- SUBSCRIPTIONS

port suggestions : (List String -> msg) -> Sub msg

subscriptions : Model -> Sub Msg
subscriptions model =
  suggestions Suggest


-- VIEW

view : Model -> Html Msg
view model =
  div []
    [ input [ onInput Change ] []
    , button [ onClick Check ] [ text "Check" ]
    , div [] [ text (String.join ", " model.suggestions) ]
    ]

Note: no word goes through port in this form, which I noticed by including a console.log(word) side effect in app.ports.check.subscribe callback. To send word across port, change update to
Check-> ( model, (check model.word) )
so check port function is called.