ERD Design for Employee Leave
To answer your questions:


Year number is part of a (three column) composite primary key.  There will be one record per employee + year + type of leave (vacation/sick/...)  Year number probably means the calendar year stored as an integer.
der_leave_cumulative_days_taken means the total of leave actually taken, which would be derived by summing the days between Employee_Leave_Taken.leave_from_date and Employee_Leave_Taken.leave_to_date where those dates fall within the year given by Employee_Annual_Leave.year_number for the same employee_id and leave_type_code.
der_leave_days_remaining is next to pointless because it would be Employee_Annual_Leave.leave_days_allowed less der_leave_cumulative_days_taken.  A real scenario would be that an employee is allowed three weeks vacation (15 days) and has taken two weeks so far (der_leave_cumulative_days_taken=10 days) and so they have 5 days remaining.  The point of keeping these precalculated in this way is that you can refer easily to these values in a query or report.  Notice the date_updated field.  This implies that there is a regular (perhaps scheduled) batch job that calculates these derived values periodically, or perhaps they are recalculated when changes are made to either leave allowed or leave taken.