Söker sammanlänkade fält

SELECT *
FROM 
    (
    SELECT `users`.`firstname`, `users`.`lastname`, `users`.`id`, 
    CONCAT(firstname," " ,lastname) AS `firstnamelastname`, 
    CONCAT(lastname," " ,firstname) AS `lastnamefirstname` 
    FROM `users` 
    WHERE (id != '15')
)
WHERE (firstnamelastname like 'a%' ) 
OR (lastnamefirstname like 'a%')

Dina alias är inte tillgängliga för MySQL när den först utvärderar WHERE klausul.

Men din fråga motsvarar faktiskt detta:

SELECT `users`.`firstname`, `users`.`lastname`, `users`.`id`, 
CONCAT(firstname," " ,lastname) AS `firstnamelastname`, 
CONCAT(lastname," " ,firstname) AS `lastnamefirstname` 
FROM `users` 
WHERE (id != '15')
AND (firstname LIKE 'a%' OR lastname LIKE 'a%')

Så din konkat i where-satsen är onödig.