Let Γ be a Q-polynomial distance-regular graph with vertex set X, diameter D ⩾ 3 and adjacency matrix A. Fix x ∈ X and let A ∗ = A ∗ ( x ) be the corresponding dual adjacency matrix. Recall that the Terwilliger algebra T = T ( x ) is the subalgebra of Mat X ( C ) generated by A and A ∗ . Let W denote a thin irreducible T-module. It is known that the action of A and A ∗ on W induces a linear algebraic object known as a Leonard pair. Over the past decade, many results have been obtained concerning Leonard pairs. In this paper, we apply these results to obtain a detailed description of W. In our description, we do not assume that the reader is familiar with Leonard pairs. Everything will be proved from the point of view of Γ . Our results are summarized as follows. Let { E i } i = 0 D be a Q-polynomial ordering of the primitive idempotents of Γ and let { E i ∗ } i = 0 D be the dual primitive idempotents of Γ with respect to x. Let r , t and d be the endpoint, dual endpoint and diameter of W, respectively. Let u and v be nonzero vectors in E t W and E r ∗ W respectively. We show that { E r + i ∗ A i v } i = 0 d and { E t + i A * i u } i = 0 d are bases for W that are orthogonal with respect to the standard Hermitian dot product. We display the matrix representations of A and A ∗ with respect to these bases. We associate with W two sequences of polynomials { p i } i = 0 d and { p i ∗ } i = 0 d . We show that for 0 ⩽ i ⩽ d , p i ( A ) v = E r + i ∗ A i v and p i ∗ ( A ∗ ) u = E t + i A * i u . Next, we show that { E r + i ∗ u } i = 0 d and { E t + i v } i = 0 d are orthogonal bases for W ; we call these the standard basis and dual standard basis for W , respectively. We display the matrix representations of A and A ∗ with respect to these bases. The entries in these matrices will play an important role in our theory. We call these the intersection numbers and dual intersection numbers of W . Using these numbers, we compute all inner products involving the standard and dual standard bases. We also use these numbers to define two normalizations u i , v i (resp. u i ∗ , v i ∗ ) for p i (resp. p i ∗ ). Using the orthogonality of the standard and dual standard bases, we show that for each of the sequences { p i } i = 0 d , { p i ∗ } i = 0 d , { u i } i = 0 d , { u i ∗ } i = 0 d , { v i } i = 0 d , { v i ∗ } i = 0 d the polynomials involved are orthogonal and we display the orthogonality relations. We also show that each of the sequences satisfy a three-term recurrence and a relation known as the Askey–Wilson duality. We then turn our attention to two more bases for W . We find the matrix representations of A and A ∗ with respect to these bases. From the entries of these matrices we obtain two sequences of scalars known as the first split sequence and second split sequence of W . We associate with W a sequence of scalars called the parameter array. This sequence consists of the eigenvalues of the restriction of A to W, the eigenvalues of the restriction of A ∗ to W , the first split sequence of W and the second split sequence of W. We express all the scalars and polynomials associated with W in terms of its parameter array. We show that the parameter array of W is determined by r , t , d and one more free parameter. From this we conclude that the isomorphism class of W is determined by these four parameters. Finally, we apply our results to the case in which Γ has q-Racah type or classical parameters.