I have changed my permalink structure to
/%category%/%postname%/.The links are working, but the problem is that my
style.css is not working in some of the posts which are inside the category.
At a guess I'd say you're linking to your stylesheet with a relative url like so:
<link href="wp-content/themes/my-theme/style.css" rel="stylesheet" />
so when you end up anywhere other than the home page it's looking at the full URL plus the path indicated to your stylesheet.
Make sure your stylesheet URL begins with a slash if you're putting it in as static HTML eg:
<link href="/wp-content/themes/my-theme/style.css" rel="stylesheet" />
Even better though is to use the built-in wordpress function for getting the stylesheet URL as Eileen has indicated in the comments. Use this syntax:
<link href="<?php bloginfo('stylesheet_url'); ?>" rel="stylesheet" />
In order to include other CSS files look into the
wp_enqueue_style() function in the codex - http://codex.wordpress.org/Function_Reference/wp_enqueue_style