The function of the word happy in the sentence is predicate adjective.
A predicate adjective is an adjective which modifies the subject and follows a linking verb. It is in contrast to an attributive adjective, which directly modifies the noun.
Therefore in the above sentence, "The band director was happy after the concert."
'Happy' is the predicate adjective which follows the linking verb 'was' and modifies the subject of the linking verb 'the band director.'