One thing that annoys me about my own A* pages is that they use grids for the examples. A* is not restricted to grids. A* works on any directed graph. A* on uniform grids is often slow, so people have come up with various ways to make the algorithm faster. I feel like the "right" thing to do is not to change the algorithm but to change the data.
Graph search is used when you want to make "global" decisions that involve potentially analyzing large parts of the map. You look ahead all the way to the end before you can decide anything. It's a waste to use it on "local" decisions that you can make without looking far ahead. Suppose you have this game map (from Baldur's Gate):
What's the easiest thing to do? Make each tile into a graph node:
This is a fine solution, but A* will take a while to find the paths. There are a lot of nodes to visit. Some algorithms will make A* faster by using a cheaper way to visit the nodes. However, it's even faster if you skip most nodes altogether. For any path on this map, almost all of steps can be decided locally, by just following a straight line. There's no need to give those nodes to A*. Instead of making every tile into a pathfinding graph node, give a smaller graph to A*:
You'll need to annotate the map with this graph, either manually in a level editor or automatically with a preprocessing algorithm. A* will run much faster on the tiny graph than the dense grid graph. If you're looking to optimize A* on a grid, consider changing the data before you consider changing the algorithm. Navigation meshes, visibility graphs, and hierarchical approaches are all worth a look.