I've this loop:

<?php while (have_posts()) : the_post(); ?>
    <li><h2 class="titles"><a href="<?php the_permalink() ?>"><?php the_title(); ?></a></h2></li>
    <li><span class="post-date">Posted <?php the_time('F j, Y'); ?> at <?php the_time('g:i a'); ?></span><span class="post-cat">Filed under: <?php the_category(', ') ?> by <?php the_author(); ?></span></li>
    <li><p><?php the_excerpt(); ?></p></li>
   <div class="voting"><?php DisplayVotes(get_the_ID()); ?></div>
<?php endwhile;?>

I simply want to put this line <div class="voting"><?php DisplayVotes(get_the_ID()); ?></div> in only when the loop is referencing a particular category, ie. if (category== 'birthdays') then output this line.

I understand how to complete an if statement but what condition will i place in?

Use get_the_category function

global $post;
$categories = get_the_category($post->ID);
var_dump($categories);

You may create custom theme for every category, Just refer to it as "category-ID.php" and edit it as you would like. Read this link http://codex.wordpress.org/Category_Templates