Making a column immutable in MySQL

Rather than abort the update, just override it?

...
FOR EACH ROW BEGIN
    SET NEW.Schueler_ID = OLD.Schueler_ID;
    SET NEW.Klasse_ID = OLD.Klasse_ID;
END

A fairly nice and clean way to abort and return a readable error message is by using a signal:

SIGNAL sqlstate '45000' SET message_text = 'Schueler_ID and Klasse_ID may not be altered!';

Do make sure your MySQL version supports this, it was introduced in MySQL 5.5. In earlier versions you need hacks like inserting a non existant field into a non existant table.

As gbn mentions, overwriting the new values with the old ones is a good option too.

I actually learned a back-alley approach to aborting triggers. It is very crude, but works because SIGNAL processing is not implemented in MySQL's Stored Procedure Language. It can be emulated at best.

Chapter 11, Pages 254-256 of the book MySQL Stored Procedure Programming under the subheading 'Validating Data with Triggers' suggests making a SELECT of a character string into a dummy integer (in terms of syntax, it is correct, but dies on execution [NO PUN INTENDED]):

CREATE TRIGGER deny_nton_change BEFORE UPDATE ON Schueler_in_Klasse 
FOR EACH ROW BEGIN
    DECLARE dummy INT; 
    IF NEW.Schueler_ID != OLD.Schueler_ID OR NEW.Klasse_ID != OLD.Klasse_ID THEN 
        SELECT 'Cannot Carry Out This Trigger Any Further'
        INTO dummy FROM table_you_know_does_not_exist WHERE 1=1 LIMIT 1;
    END IF; 
END;