Find the Simple Cycles in a Directed Graph
This question appears to be very Googleable. For example, you may be interested in the algorithm presented in this paper:

Finding all the elementary circuits of a directed graph. Donald B. Johnson. SIAM J. COMPUT. Vol. 4, No. 1, March 1975


  Abstract. An algorithm is presented which finds all the elementary circuits-of a directed graph in time bounded by O((n + e)(c + 1)) and space bounded by O(n + e), where there are n vertices, e edges and c elementary circuits in the graph. The algorithm resembles algorithms by Tiernan and Tarjan, but is faster because it considers each edge at most twice between any one circuit and the next in the output sequence.


The paper contains a complete algorithm.