I have a wordpress based website and the default category permalink is like "website.com/category/category-slug"
How can I make it look like "website.com/category-slug"? Of course I will always take care that there is no post/page with the same slug
This should work, although not tested. Fill in the categoryID below and set your desired URL in wp_redirect().
<?php
$category_id = xx;
$category_link = get_category_link( $category_id )
wp_redirect( $category_link, 'http://www.website.com/category-slug' ); exit;
?>
Remember by the way that wp_redirect() always should be followed by exit; (Source: Codex).