We obtain an explicit criterion for $p$-ary functions to produce association schemes in terms of their Walsh spectrum. Employing this characterization, we explicitly find a correlation between $p$-ary bent functions and association schemes; to be more exact, we prove that a $p$-ary bent function induces a $p$-class association scheme if and only if the function is weakly regular. As applications of our main criterion, we construct many infinite families of few-class association schemes arising from $p$-ary functions. Furthermore, we present four classes of $p$-ary two-weight linear codes, which are constructed from the association schemes produced in this paper.