Select which has max date or latest date

Your current query is not giving the desired result because you are using a GROUP BY clause on the PERSON_ID column which has a unique value for both entries. As a result you will return both rows.

There are a few ways that you can solve this. You can use a subquery to apply the aggregate function to return the max(LAST_UPDATE_DATE_TIME) for each SCHOOL_CODE:

select s1.LAST_UPDATE_DATE_TIME,
  s1.SCHOOL_CODE,
  s1.PERSON_ID
from SCHOOL_STAFF s1
inner join
(
  select max(LAST_UPDATE_DATE_TIME) LAST_UPDATE_DATE_TIME,
    SCHOOL_CODE
  from SCHOOL_STAFF
  group by SCHOOL_CODE
) s2
  on s1.SCHOOL_CODE = s2.SCHOOL_CODE
  and s1.LAST_UPDATE_DATE_TIME = s2.LAST_UPDATE_DATE_TIME;

See SQL Fiddle with Demo

Or you can use use a windowing function to return the rows of data for each school with the most recent LAST_UPDATE_DATE_TIME:

select SCHOOL_CODE, PERSON_ID, LAST_UPDATE_DATE_TIME
from
(
  select SCHOOL_CODE, PERSON_ID, LAST_UPDATE_DATE_TIME,
    row_number() over(partition by SCHOOL_CODE 
                        order by LAST_UPDATE_DATE_TIME desc) seq
  from SCHOOL_STAFF
  where STAFF_TYPE_NAME='Principal'
) d
where seq = 1;

See SQL Fiddle with Demo

This query implements row_number() which assigns a unique number to each row in the partition of SCHOOL_CODE and placed in a descending order based on the LAST_UPDATE_DATE_TIME.

As a side note, the JOIN with aggregate function is not exactly the same as the row_number() version. If you have two rows with the same event time the JOIN will return both rows, while the row_number() will only return one. If you want to return both with a windowing function, then consider using the rank() windowing function instead as it will return ties:

select SCHOOL_CODE, PERSON_ID, LAST_UPDATE_DATE_TIME
from
(
  select SCHOOL_CODE, PERSON_ID, LAST_UPDATE_DATE_TIME,
    rank() over(partition by SCHOOL_CODE 
                        order by LAST_UPDATE_DATE_TIME desc) seq
  from SCHOOL_STAFF
  where STAFF_TYPE_NAME='Principal'
) d
where seq = 1;

See Demo

I'm surprised nobody has taken advantage of window functions beyond row_number()

Here's some data to play with:

CREATE TABLE SCHOOL_STAFF
(
LAST_UPDATE_DATE_TIME VARCHAR(20),
SCHOOL_CODE VARCHAR(20),
PERSON_ID VARCHAR(20),
STAFF_TYPE_NAME VARCHAR(20)
);
INSERT INTO SCHOOL_STAFF VALUES ('24-JAN-13', 'ABE', '111222', 'Principal');
INSERT INTO SCHOOL_STAFF VALUES ('09-FEB-12', 'ABE', '222111', 'Principal');

The OVER() clause creates a window for which you will define your aggregate groups. In this case, I am only partitioning on the SHOOL_CODE, so we will see the FIRST_VALUE, which will come from LAST_UPDATE_DATE_TIME, grouped by SCHOOL_CODE, and in the order of LAST_UPDATE_DATE_TIME by descending order. This value will be applied to the entire column for each SCHOOL_CODE.

It is important to pay close attention to your partitioning and ordering in the over() clause.

SELECT DISTINCT
 FIRST_VALUE(LAST_UPDATE_DATE_TIME) OVER (PARTITION BY SCHOOL_CODE ORDER BY LAST_UPDATE_DATE_TIME DESC) AS LAST_UPDATE
,FIRST_VALUE(SCHOOL_CODE)           OVER (PARTITION BY SCHOOL_CODE ORDER BY LAST_UPDATE_DATE_TIME DESC) AS SCHOOL_CODE
,FIRST_VALUE(PERSON_ID)             OVER (PARTITION BY SCHOOL_CODE ORDER BY LAST_UPDATE_DATE_TIME DESC) AS PERSON_ID
FROM SCHOOL_STAFF
WHERE STAFF_TYPE_NAME = 'Principal'
ORDER BY SCHOOL_CODE

Returns:

24-JAN-13   ABE 111222

This should eliminate your need for GROUP BY and Subqueries for the most part. You will want to make sure to include DISTINCT though.