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."
<em>'Happy'</em> is the predicate adjective which follows the linking verb <em>'was</em>' and modifies the subject of the linking verb <em>'the band director.'</em>
