You need to add a "LIMIT" clause to your original query. For instance,
$result = mysql_query("SELECT * FROM berichten ORDER BY id DESC LIMIT 10, 10")
Will skip over the first 10 results, so the first result you get will be the 11th, and the last the 20th.
You will want to develop this further using a constant for the number of items per page and querying some kind of variable, most likely in the $_GET array, for how many items to skip over:
// number of items to show on each page
define('ITEMS_PER_PAGE', 10);
// now we're getting the number to start fetching from. This line of code is saying,
// if $_GET['start'] is set and it is numeric, then use it, otherwise default to zero
$start = (isset($_GET['start']) && is_numeric($_GET['start'])) ? $_GET['start'] : 0;
// now do the query: notice now that I'm actually fetching one more than the
// ITEMS_PER_PAGE constant: this will allow us to know whether or not to show
// another "next 10 items" link
$result = mysql_query("SELECT * FROM berichten ORDER BY id DESC LIMIT $start, " . ITEMS_PER_PAGE + 1)
I'm running out of time now but then you can go ahead, iterate through and print out a link which will be something like:
echo '<a href="yourpage.php?start=' . $start + 10 . '">next 10 »</a>';
I personally prefer a "get next 10 items" link which uses ajax and similar techniques to get the next 10.
*disclaimer: above isn't debugged!