Finding the k-shortest path between two nodes
In the $k$ shortest path problem, we wish to find $k$ path connecting a given vertex pair with minimum total length. Eppstein [1] has an algorithm running in $O(m+n \log n + k)$ time to find the $k$ shortest paths (allowing cycles) between a pair of vertices in a digraph. With the techniques of the paper, one can also find all path shorter than some given threshold, within the same time bounds. There is a vast literature on the subject, the Eppstein paper includes many references and discussion.

If you disallow cycles, you might want to look at the algorithm of Hershberger et al. [2]. 



[1] Eppstein, David. "Finding the k shortest paths." SIAM Journal on computing 28.2 (1998): 652-673. [CiteSeerX]

[2] Hershberger, John, Matthew Maxel, and Subhash Suri. "Finding the k shortest simple paths: A new algorithm and its implementation." ACM Transactions on Algorithms (TALG) 3.4 (2007): 45. [CiteSeerX]