Wordpress - How can you check if you are in a particular page in the WP Admin section? For example how can I check if I am in the Users > Your Profile page?

There is a global variable in wp-admin called $pagenow which holds name of the current page, ie edit.php, post.php, etc.

You can also check the $_GET request to narrow your location down further, for example:

global $pagenow;
if (( $pagenow == 'post.php' ) && ($_GET['post_type'] == 'page')) {

    // editing a page

    }

if ($pagenow == 'users.php') {

    // user listing page

    }

if ($pagenow == 'profile.php') {

    // editing user profile page

    }

The way to do this is to use the 'admin_enqueue_scripts' hook to en-queue the files you need. This hook will get passed a $hook_suffix that relates to the current page that is loaded:

function my_admin_enqueue($hook_suffix) {
    if($hook_suffix == 'appearance_page_theme-options') {
        wp_enqueue_script('my-theme-settings', get_template_directory_uri() . '/js/theme-settings.js', array('jquery'));
        wp_enqueue_style('my-theme-settings', get_template_directory_uri() . '/styles/theme-settings.css');
        ?>
        <script type="text/javascript">
        //<![CDATA[
        var template_directory = '<?php echo get_template_directory_uri() ?>';
        //]]>
        </script>
        <?php
    }
}
add_action('admin_enqueue_scripts', 'my_admin_enqueue');

The most comprehensive method is get_current_screen added in WordPress 3.1

$screen = get_current_screen();

returns

WP_Screen Object (
    [action] => 
    [base] => post
    [id] => post
    [is_network] => 
    [is_user] => 
    [parent_base] => edit
    [parent_file] => edit.php
    [post_type] => post
    [taxonomy] => 
)