Applications of Kruskal and Prim's algorithms

Minimum spanning trees were first studied for ways to lay out electrical networks in a way that minimizes the total cost of the wiring. In a minimum spanning tree, all the nodes (houses) would be connected to power by wires in a way that has minimum cost and redundancy (cutting any wire necessarily cuts the power grid into two pieces).

Since then, the problem has been well-studied and is often used as a subroutine in more complex algorithms. The Christofides algorithm for finding approximate solutions to the Traveling Salesman Problem uses it in a key step, as do some algorithms for finding Steiner trees.

Minimum spanning trees have also been used to generate mazes. Both Kruskal's and Prim's algorithm have been used this way, often creating high-quality mazes.

If you're interested in a full history of the minimum spanning tree problem, its applications, and its algorithms, there is a truly excellent paper available here that covers all of these. I'd strongly suggest giving it a read!

Hope this helps!

Quoting Wikipedia:

One example would be a cable TV company laying cable to a new neighborhood. If it is constrained to bury the cable only along certain paths, then there would be a graph representing which points are connected by those paths. Some of those paths might be more expensive, because they are longer, or require the cable to be buried deeper; these paths would be represented by edges with larger weights. A spanning tree for that graph would be a subset of those paths that has no cycles but still connects to every house. There might be several spanning trees possible. A minimum spanning tree would be one with the lowest total cost.

Source: http://en.wikipedia.org/wiki/Minimum_spanning_tree