Richard Stanley has written some excellent notes in EC2 ch. 5 p67 about the Lagrange inversion formula (Theorem 5.4.2 on p38).
Stanley provides reference to An introduction to the theory of infinite series by Thomas John I'Anson Bromwich (You can see the full 1908 text through google books and other editions here: http://books.google.com/books?q=editions:UOM39015064521290&id=ZY45AAAAMAAJ) whereseveral where several applications are provided.
Stanley also provides reference for generalizations of the Lagrange inversion formula:
I.M. Gessel's Paper (combinatorial proof): http://portal.acm.org/citation.cfm?id=31572 Note: Gessel also gives a generalization of Lagrange inversion to noncommutative power series in A noncommutative generalization and q-analog of the Lagrange inversion formula.
See also D.W. Stanton's Survey: Recent results for the q-Lagrange inversion formula.
