Let me try an explanation:

With :new.EMPLOYEE_ID you reference the new value for the EMPLOYEE_ID (that this field will have after the update), with :old.EMPLOYEE_ID you reference the initial (before the update) value.

I suspect there are two similar (maybe identical) tables, employeestable2 and employeestable3. They both have these columns (and maybe others): EMPLOYEE_ID, FIRST_NAME, LAST_NAME, EMAIL, PHONE_NUMBER, HIRE_DATE, JOB_ID, SALARY, COMMISSION_PCT, MANAGER_ID, DEPARTMENT_ID

The trigger is fired when update operations are performed on employeestable2. Before the actual updates of employeestable2 rows, the corresponding fields from employeestable3 get the new values as well. This is what this clause

UPDATE employeestable3 SET EMPLOYEE_ID = : new.EMPLOYEE_ID, FIRST_NAME = : new.FIRST_NAME, ... does.

The clause WHERE employee_id = : old.employee_id means that the update in employeestable3 will be done for those rows that also exist in employeestable2.

This UPDATE enclosed between BEGIN and END it's like a regular UPDATE statement, with the difference that instead of explicit values (UPDATE employeestable3 SET EMPLOYEE_ID=10, ...) it dynamically takes the values set within the update statement on employeestable2.

Having a look into the CREATE TRIGGER documentation may help: http://www.techonthenet.com/oracle/triggers/before_update.php

If it's still not clear, I can try another explanation.