Dijkstra's algorithm finds the shortest path between two nodes in a graph. It works by tracking the cost of the shortest path from the starting node to all other visited nodes. At each step, it visits the unvisited node with the lowest cost and updates the costs of paths through that node. The algorithm terminates when it has visited the destination node and returns the final cost of the path from start to finish. The example applies Dijkstra's algorithm to a weighted graph to find the lowest cost path from node a to z, determining the cost is 7 units.