Let \(\Lambda\) be an algebra over the power series ring \(T=k[[x,y]]\), \(k\) being algebraically closed with \(\operatorname{char} k = 0\), and \(\Lambda\) finitely generated and free as a \(T\)-module. \(\Lambda\) is said to be representation-finite if the category of finitely generated reflexive \(\Lambda\)-modules has a finite number of indecomposables. The representation-finite algebras \(\Lambda\) are characterized in terms of their Auslander-Reiten quivers. If \(\Lambda\) is representation-finite and indecomposable, then \(\Lambda\) is a tame order in a simple algebra. The Grothendieck group \(G\) of the Auslander-Reiten quiver of \(\Lambda\) is defined, and it is proved that the length of a maximal chain of tame overorders of \(\Lambda\) is just the rank of \(G\). Thus, a criterion for maximal orders \(\Lambda\) is derived, and in particular, this yields a new proof of M. Artin’s classification of representation-finite maximal orders in division rings [Invent. Math. 84, 195–222 (1986; Zbl 0591.16002); Manuscr. Math. 58, 445–471 (1987; Zbl 0625.16005)].