How to display a “NEW”-icon on posts from today

you can use the difference between the post time (get_the_time()) and the current time (date()) –

for instance, this snippet added to functions.php of your theme would add a new css class to post_class() if the post is less than 24hrs old:

function post_from_today_class($class) {
//add .new-post-today to post_class() if newer than 24hrs
    global $post;
    if( date('U') - get_the_time('U', $post->ID) < 24*60*60 ) $class[] = 'new-post-today';
    return $class;
}

add_filter('post_class','post_from_today_class');

you can use the .new-post-today class to style something differently in your post div.

the core idea is:

if( date('U') - get_the_time('U', $post->ID) < 24*60*60 ) { /*do something*/ };

edit:
alternatively, direct in a template with its own php tags, use:

<?php if( date('U') - get_the_time('U', $post->ID) < 24*60*60 ) : ?>text<?php endif; ?>