Loop no.1

<?php
$do_not_duplicate = array(); // set befor loop variable as array

// 1. Loop
query_posts('ca=1,2,3&showposts=5');
while ( have_posts() ) : the_post();
    $do_not_duplicate[] = $post->ID; // remember ID's in loop
    // display post ...
    the_title();
endwhile;
?>

After we output the above loop with 5 articles, we have in the array 5 IDs, which we can use now . You can view the array with the PHP-function var_dump().

In the next loop no. 2 there should be 15 articles displayed, except the articles with the ID's already appearing in loop no. 1. I check with this function in_array(),
if the current ID $post->ID is already existing in the array $do_not_duplicate. Only if the ID is not existing ( !in_array() ) in the array, then it will be displayed in loop no. 2.

Loop no.2

<?php
// 2. Loop
query_posts( 'cat=4,5,6&showposts=15' );
while (have_posts()) : the_post();
    if ( !in_array( $post->ID, $do_not_duplicate ) ) { // check IDs
// display posts ...
        the_title();
    }
endwhile;
?>

But there is also an alternative, WordPress has a parameter in the query available - post__not_in, see Codex.
Also in this parameter, I use an array and in this query are the IDs not included. Depending on which way you go, are thus provides two different syntax, to avoid duplicate content.

<?php
// another loop without duplicates
query_posts( array(
    'cat' => 456,
    'post__not_in' => $do_not_duplicate
    )
);
while ( have_posts() ) : the_post();
    // display posts...
        the_title();
endwhile;
?>

if ( !function_exists('fb_filter_child_cats') ) {
	function fb_filter_child_cats( $cats ) {
		global $wp_query, $wpdb;

		if ( is_category() ) {

			// get children ID's
			if ( $excludes = get_categories( "child_of=" . $wp_query->get('cat') ) ) {

				// set array with ID's
				foreach ( $excludes as $key => $value ) {
					$exclude[] = $value->cat_ID;
				}
			}

			// remove child cats
			if ( isset($exclude) && is_array($exclude) ) {
				$cats .= " AND " . $wpdb->prefix . "term_taxonomy.term_id NOT IN (" . implode(",", $exclude) . ") ";
			}
		}

		return $cats;
	}

	if ( !is_admin() ) {
		add_filter( 'posts_where', 'fb_filter_child_cats' );
	}
}

Update for WP 3.1 and higher:

if ( !function_exists('fb_filter_child_cats') ) {
	function fb_filter_child_cats( $cats ) {
		global $wp_query, $wpdb;
		if ( is_category() ) {
			// get children ID's
			if ( $excludes = get_categories( "child_of=" . $wp_query->get('cat') ) ) {
				// set array with ID's
				foreach ( $excludes as $key => $value ) {
					$exclude[] = $value->cat_ID;
				}
			}
			// remove child cats
			if ( isset($exclude) && is_array($exclude) ) {
				$cats .= " AND " . $wpdb->prefix . "term_relationships.term_taxonomy_id NOT IN (" . implode(",", $exclude) . ") ";
			}
		}
		return $cats;
	}
	if ( !is_admin() ) {
		add_filter( 'posts_where', 'fb_filter_child_cats' );
	}
}

and also Update 3 for a better solution without an sql-select:

function fb_filter_child_cats($query) {

	$cat = get_term_by('name', $query->query_vars['category_name'], 'category');
	$child_cats = (array) get_term_children( &$cat->term_id, 'category' );
	// also possible
	// $child_cats = (array) get_term_children( get_cat_id($query->query_vars['category_name']), 'category' );

	if ( !$query->is_admin )
		$query->set( 'category__not_in', array_merge($child_cats) );

	return $query;
}
add_filter( 'pre_get_posts', 'fb_filter_child_cats' );

I will show some other examples, which are barely used, but often asked by users, in this post. Short and sweet, learn and use.

Page break with wp-query

If you use own loops, which are defined through the variable $wp_query, created by WP-Query, then it won't display any pagination. Two examples with different ways to approach.

<?php if (have_posts()) :
	$my_query = new WP_Query('cat=-21&showposts=3'.'&paged='.$paged);
	while ( $my_query->have_posts() ) : $my_query->the_post();
?>

<?php if (have_posts()) : ?>
     <?php $paged = (get_query_var('paged')) ? get_query_var('paged') : 1;
            query_posts("category_name=somecat&paged=$paged"); ?>
        <?php while (have_posts()) : the_post(); ?>

Everybody who likes to create a special case with the loop, should check out the array of this variable more closely. Because many useful possibilities are available and can replace other solutions. For example this one with a simple syntax:

var_dump( $wp_query );

For PHP users nothing new, but maybe for some theme authors.

Exclude categories in a loop

Very often you like to exclude a category in a loop. A typical example is, if you like to have a side- or tumbler-blog within your WordPress installation. You don't need a Plugin for that. You can select a category, which will be displayed via extra loop in your sidebar, to display short tips and tricks. In your main blog you just exclude this category, so only posts with more content will be listed.

<?php query_posts('cat=-2'); ?>

A standard loop, which excludes category with ID 21

The easiest way to build a loop, take out the category and so the pagination. Below you see a loop, where you just have to fill in some content.

<?php if ( have_posts() ) :
	$my_query = new WP_Query('cat=-21');
	while ( $my_query->have_posts() ) : $my_query->the_post();
?>
		<div id="post-<?php the_ID(); ?>">
			. . .
		</div>
	<?php endwhile; else: ?>
	<p><?php _e('Not found, what you are looking for.'); ?></p>
<?php endif; ?>

Own loop, which defines paged

<?php
$paged = (get_query_var('paged')) ? get_query_var('paged') : 1;
query_posts($query_string . "&cat=-2&paged=$paged");
?>

Reset query

Problems with your own query? If you use several loops and there is an error coming up, it helps to reset. Not much documented but in my opinion very helpful. I use this, if I load several queries or have a problem in the output.

Putting it before or after a query query_posts(), works like magic sometimes.

/**
 * Destroy the previous query and setup a new query.
 *
 * This should be used after {@link query_posts()} and before another {@link
 * query_posts()}. This will remove obscure bugs that occur when the previous
 * wp_query object is not destroyed properly before another is setup.
 *
 * @since 2.3.0
 * @uses $wp_query
 */
wp_reset_query()

A small example for that. The first loop gives you the number of posts back, which contains the search. After that I put a reset and continue with standard loop to display the outcome. I wrote an article for the outcome of a search.

<?php if (have_posts()) : ?>

	<h2>Search Results<br /><small>
	<?php /* Search Count */
		$allsearch =& new WP_Query("s=$s&showposts=-1");
		$key = wp_specialchars($s, 1);
		$count = $allsearch->post_count;
		_e('term: ');
		echo $key;
		_e(' — ');
		echo $count . ' ';
		_e('posts'); wp_reset_query(); ?></small></h2>

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