Here is the code I use in a WordPress site’s index.php file to exclude a specific category from the home page withOUT breaking the pagination:
<?php if ( is_home() ) { query_posts( $query_string.'&cat=-21' ); } ?>
Of course, to include a specific category, you would use this:
<?php if ( is_home() ) { query_posts( $query_string.'&cat=4' ); } ?>
Here is the code I use to show a certain number of the most recent posts including their thumbnails from specific categories in a WordPress website.
To show the 3 most recent posts from Category 1, you would add this code to the appropriate template file of your site:
<?php
//for category 1, show 3 posts
$cat_args=array(
'include' => '1',
'orderby' => 'name',
'order' => 'ASC'
);
$categories=get_categories($cat_args);
foreach($categories as $category) {
$args=array(
'showposts' =>3,
'category__in' => array($category->term_id),
'caller_get_posts'=>1
);
$posts=get_posts($args);
if ($posts) {
echo '<p><h4>Latest <a href="' . get_category_link( $category->term_id ) . '" title="' . sprintf( __( "View all posts in %s" ), $category->name ) . '" ' . '>' . $category->name.' Posts</a>: </h4></p> ';
foreach($posts as $post) {
setup_postdata($post); ?>
<p class="recent-posts"><span class="thumb"><?php if ( has_post_thumbnail() ) { the_post_thumbnail(); } ?></span>
<span class="recent-post"><a href="<?php the_permalink() ?>" rel="bookmark" title="Permanent Link to <?php the_title_attribute(); ?>"><?php the_title(); ?></a></span></p>
<?php
} // foreach($posts
} // if ($posts
} // foreach($categories
?>
You will want to edit the category ID(s) in this code of course so that you call the posts from the correct categories of your site, and you may also want to edit the number of posts shown from the categories as well.
To show the 3 most recent posts from Categories 1, 2, AND 3, you would use this code:
<?php
//for categories 1, 2, and 3, show 3 posts
$cat_args=array(
'include' => '1,2,3',
'orderby' => 'name',
'order' => 'ASC'
);
$categories=get_categories($cat_args);
foreach($categories as $category) {
$args=array(
'showposts' =>3,
'category__in' => array($category->term_id),
'caller_get_posts'=>1
);
$posts=get_posts($args);
if ($posts) {
echo '<p><h4>Latest <a href="' . get_category_link( $category->term_id ) . '" title="' . sprintf( __( "View all posts in %s" ), $category->name ) . '" ' . '>' . $category->name.' Posts</a>: </h4></p> ';
foreach($posts as $post) {
setup_postdata($post); ?>
<p class="recent-posts"><span class="thumb"><?php if ( has_post_thumbnail() ) { the_post_thumbnail(); } ?></span>
<span class="recent-post"><a href="<?php the_permalink() ?>" rel="bookmark" title="Permanent Link to <?php the_title_attribute(); ?>"><?php the_title(); ?></a></span></p>
<?php
} // foreach($posts
} // if ($posts
} // foreach($categories
?>
If you want to include the excerpt of each post, you would use this code:
<?php
//for category 1, show 3 posts
$cat_args=array(
'include' => '1',
'orderby' => 'name',
'order' => 'ASC'
);
$categories=get_categories($cat_args);
foreach($categories as $category) {
$args=array(
'showposts' =>3,
'category__in' => array($category->term_id),
'caller_get_posts'=>1
);
$posts=get_posts($args);
if ($posts) {
echo '<p><h4>Latest <a href="' . get_category_link( $category->term_id ) . '" title="' . sprintf( __( "View all posts in %s" ), $category->name ) . '" ' . '>' . $category->name.' Posts</a>: </h4></p> ';
foreach($posts as $post) {
setup_postdata($post); ?>
<p class="recent-posts"><span class="thumb"><?php if ( has_post_thumbnail() ) { the_post_thumbnail(); } ?></span>
<span class="recent-post"><a href="<?php the_permalink() ?>" rel="bookmark" title="Permanent Link to <?php the_title_attribute(); ?>"><?php the_title(); ?></a></span>
<br/><span class="excerpt"><?php the_excerpt(); ?></span></p>
<?php
} // foreach($posts
} // if ($posts
} // foreach($categories
?>
Let me know if you have any questions or need any further modifications to this code 
Here is the code I use to make sure the embedded YouTube videos on a website don’t appear on top of other content (for example: drop down menus):
<object width='400' height='350'> <param name='movie' value='http://www.youtube.com/YOURVIDEOLINK'> <param name='type' value='application/x-shockwave-flash'> <param name='allowfullscreen' value='true'> <param name='allowscriptaccess' value='always'> <param name="wmode" value="opaque" /> <embed width='400' height='350' src='http://www.youtube.com/YOURVDIDEOLINK' type='application/x-shockwave-flash' allowfullscreen='true' allowscriptaccess='always' wmode="opaque" ></embed> </object>
The areas of code highlighted in red above are what solve the overlapping/z-index problem, so if you already have your video embedded on your site, simply add those sections in the right place in your code and that should solve the issue.
NOTE: I found this solution at http://australiansearchengine.wordpress.com/2010/06/19/youtube-video-css-z-index/
Here is a simple block of code that you can use to get the most recent post titles and permalinks to a certain number of the most recent posts in a certain category:
<ul>
<?php
global $cat;
$recentPosts = new WP_Query();
$recentPosts->query('showposts=5&cat=389');
?>
<?php while ($recentPosts->have_posts()) : $recentPosts->the_post(); ?>
<li><a href="<?php the_permalink() ?>" rel="bookmark"><?php the_title(); ?></a></li>
<?php endwhile; ?></ul>
Simply change the number of posts and the category id in the code above to get the number of posts and the category of posts that you want.
You can also check out my post on how to display a list of the titles of your most recent blog posts from several categories if you are looking for a way to get the most recent posts from several different categories.