I've four groups (That like, recent, popular, all) with 16 posts in every. So, for that That like i must show the 15 in the That like category and the other way around for that recent page etc.

Post 1 Post 2 Post 3 Post 4
Post 5 Post 6 Post 7 Post 8
Post 9 Post 10 Post 11 Post 12
Post 13 Post 14 Post 15 Post 16

You should use using the category id:

$query = new WP_Query( 'cat=4' );

or while using category title (but the first is much better)

$query = new WP_Query( 'category_name=staff' );

$query provides you with their email list of posts (filters applied)

The limit could be defined with "publish_count" and also the position sohuld be based on CSS (float:left possibly). More details about Wordpress_Query on http://codex.wordpress.org/Function_Reference/WP_Query