There were few syntax errors in your HTML and JavaScript code. Please try below:

HTML:

<p>
<label for="method">How?</label>
<select id="method" name="method" required>
    <option value="email">Email</option>
    <option value="web">Web</option>
</select> <span class="help-text">The web, or email?</span>
</p>
<p>
<label id="contact-label" for="contact">Email Address</label>
<input type="email" name="contact" id="contact" value="" maxlength="255"> <span class="help-text">e.g. stuff@yourdomain.com</span>
    </p>

JavaScript:

$(document).ready(function(){
    $('#method').change(
        function () {
            var method = $('option:selected', this).text();
            if (method == "Email") {
                $('#contact-label').text("Email Address");
            } else if (method == "Web") {
                $('#contact-label').html("App URL");
            }
        });
 });  

try something like this,FIDDLE

$(document).ready(function () {
    $('#method').change(
    function () {
        var method = $('option:selected').val();

        if (this.value == "email") {
            $('#contact').text("Email Address");
        } else if (this.value == "web") {
            $('#contact').text("Web Address");
        }
    });

});

Check this fidle

        $(document).ready(

    $('#method').bind('change',
      function(){
      var method = $('option:selected',this).text();
      alert(method);
      if (method == "Email") {
          $('#contact-label').text("Email Address"); 
     $('#help-change').text("Email Address"); 
          $('#contact').attr('type','email');

      } else if (method == "Web") {
          $('#contact-label').text("App URL"); 
          $('#help-change').text("Web Address"); 
          $('#contact').attr('type','url');
      }

      }
      )

  );

http://jsfiddle.net/ys9Cj/18/

$("#method").change(function () {
    $('#contact').html(($(this).find(':selected').html() == 'Email' ? 'Email Address' : 'Web Address') + ':');       
});