ajax.php

<?php // Load items into a HTML select box via PHP & Ajax

$firstbox = $_POST['firstbox'];
$query = "SELECT * FROM mytable WHERE id_firstbox = '$firstbox';";
$result = pg_query($query) or die('Query failed: ' . pg_last_error());
$secondbox_arr = array();
if (pg_num_rows($result) > 0) {
  while ($row = pg_fetch_array($result, NULL, PGSQL_ASSOC)) {
    $secondbox_arr[] = array("id" => $row['id_secondbox'], "name" => $row['testo_secondbox']);
  }
}
echo json_encode($secondbox_arr);

index.php

<form>
  <label>First Box:
    <select id="firstbox" name="firstbox_name">
      <option value=" "></option>
      <option value="1">Option 1</option>
      <option value="2">Option 2</option>
      <option value="3">Option 3</option>
    </select>
  </label>

  <label>Second Box:
    <select id="secondbox" name="secondbox_name">
      <option value=" "></option>
    </select>
  </label>
</form>

<script>
  $().ready(function() {
    var $secondbox = $("#secondbox");
    $("#firstbox").change(function() {
      var firstbox_id = $(this).val();
      $secondbox.empty();
      if (this.value.trim() !== '') {
        $.ajax({
          url: 'ajax.php',
          type: 'post',
          data: {firstbox: firstbox_id},
          dataType: 'json',
          success: function (response) {
            var len = response.length;
            for (var i = 0; i < len; i++) {
              var id = response[i]['id'];
              var name = response[i]['name'];
              $name.append("<option value='" + id + "'>" + name + "</option>");
            }
          },
          error: function (exception) {
            alert('Exception: ' + exception);
          }
        });
      }
    });
  });
</script>