How do I make the category template display full posts instead of partial posts?

@Ben,

If you look at line 132 of loop.php in Twentyten you will find the conditional statement that is telling WordPress to only display the_excerpt on category archives. Right below that after <?php else : ?> is how all posts that are not on an archive page, in the asides category or in the gallery category will be displayed. (The asides and gallery categories are targeted at the beginning of loop.php)

<?php if ( is_archive() || is_search() ) : // Only display excerpts for archives and search. ?>
            <div class="entry-summary">
                <?php the_excerpt(); ?>
            </div><!-- .entry-summary -->
    <?php else : ?>
            <div class="entry-content">
                <?php the_content( __( 'Continue reading <span class="meta-nav">&rarr;</span>', 'twentyten' ) ); ?>
                <?php wp_link_pages( array( 'before' => '<div class="page-link">' . __( 'Pages:', 'twentyten' ), 'after' => '</div>' ) ); ?>
            </div><!-- .entry-content -->
    <?php endif; ?>

In your loop-category.php file the easiest thing to do would be copy and paste the whole loop.php file and change the section of code above to:

<?php if ( is_search() ) : // Only display excerpts for archives and search. ?>
            <div class="entry-summary">
                <?php the_excerpt(); ?>
            </div><!-- .entry-summary -->
    <?php else : ?>
            <div class="entry-content">
                <?php the_content( __( 'Continue reading <span class="meta-nav">&rarr;</span>', 'twentyten' ) ); ?>
                <?php wp_link_pages( array( 'before' => '<div class="page-link">' . __( 'Pages:', 'twentyten' ), 'after' => '</div>' ) ); ?>
            </div><!-- .entry-content -->
    <?php endif; ?>

You will notice that we only removed is_archive from the conditional statement so now category archives will displayed using the code after the else statement which contains the_content as opposed to the_excerpt.