I will be summarizing my comments so you can mark this post here as solution for the next person having your problem. This makes finding the answer easier somehow
You need a webserver which supports rewriting of URLs. Most people perhaps use the apache2 webserver which does supports this. The module for apache doing this is called "mod_rewrite". You probably (depending on your configuration) need to enable it first.
Clean and beautiful URLs are called "smart-urls", for this term makes searching for tutorials, guides and answers much easier.
To make the mod_rewrite work and rewrite your URLs you need to enable the module for your current project/directory and write some rules down.
You can do this within your .htaccess file and for your example it will look something like this:
.htaccess:
RewriteEngine On
RewriteRule ^news/([0-9]+)\.?.*$ news.php?id=$1 [NC,L]
This will cause the server internally (and without the user can see) to rewrite something like this:
http://www.foo.de/news/1337.my-awesome-first-newspost
to
http://www.foo.de/news.php?id=1337
Remark: Everything behind the News ID is ignored
As you can see rules are written as regular expressions which enables you to create really flexible rules.
You can also write multiple rules which depend on each other or just become applied one after the other.
I found a page enabling you to test your rules because this is always a bit hard "debugging" when doing it on your server machine:
http://htaccess.madewithlove.be/