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The category 1 above didn't display corrected on here, but my example category 2 and 3 is how I would like it to be.

<?php
//get all terms (e.g. categories or post tags), then display all posts in each term
$taxonomy = 'category';//  e.g. post_tag, category
$param_type = 'category__in'; //  e.g. tag__in, category__in
$term_args=array(
  'orderby' => 'name',
  'order' => 'ASC',
);
$terms = get_terms($taxonomy,$term_args);
if ($terms) {
  foreach( $terms as $term ) {
    $args=array(
      "$param_type" => array($term->term_id),
      'post_type' => 'post',
      'post_status' => 'publish',
      'posts_per_page' => -1,
      'caller_get_posts'=> 1
      );
    $my_query = null;
    $my_query = new WP_Query($args);
    if( $my_query->have_posts() ) {
      echo 'List of Posts in '.$taxonomy .' '.$term->name;
      while ($my_query->have_posts()) : $my_query->the_post(); ?>
        <p><a href="<?php the_permalink() ?>" rel="bookmark" title="Permanent Link to <?php the_title_attribute(); ?>"><?php the_title(); ?></a></p>
       <?php
      endwhile;
    }
  }
}
wp_reset_query();  // Restore global post data stomped by the_post().
?>

Thanks Michael! How can I exclude specific categories from this? Thanks again in advance.

something like this:

<ul class="cat_list">
<?php  $cats = get_categories('orderby=count&order=DESC');
// get all categories;
foreach ($cats as $cat) :

echo '<li class="cat_list_item">';
echo '<h2>'.$cat->name.'</h2>'; 

$args = array(
'posts_per_page' => -1, // max number of post per category
'cat' => $cat->term_id
);
query_posts($args); // get all posts for category

if (have_posts()) :
echo '<ul class="post_list">'; //start the unordered list of posts

while (have_posts()) : the_post(); ?>	

  <li>
  <a href="<?php the_permalink() ?>" rel="bookmark" title="Permanent Link to <?php the_title_attribute(); ?>"><?php the_title(); ?></a>
  </li>

<?php endwhile; 

echo '</ul>'; //end of unordered post list
endif; 

wp_reset_query(); ?>	

</li> <!--end of li cat_list_item -->
<?php endforeach; ?>
</ul> <!-- end of unordered cat list

untested

Thanks for that alchymyth, Michael's solution did exactly what I needed, now I just need the ability to exclude certain categories from it.

Change this

$term_args=array(
  'orderby' => 'name',
  'order' => 'ASC',
);

to something like this (you supply the correc ids to exclude):

$term_args=array(
  'exclude' => '1,2,3',
  'orderby' => 'name',
  'order' => 'ASC'
);

That worked perfectly. Thanks so much!!