A* path finding is a best-first type search that uses an additional heuristic.
The first thing you need to do is divide up your search area. For this explanation the map is a square grid of tiles, because most 2D games use a grid of tiles and because that's simple to visualize. Note however that the search area can be broken up in any way you want: a hex grid perhaps, or even arbitrary shapes like Risk. The various map positions are referred to as "nodes" and this algorithm will work any time you have a bunch of nodes to traverse and have defined connections between the nodes.
Anyway, starting at a given starting tile:
The 8 tiles around the starting tile are "scored" based on a) the cost of moving from the current tile to the next tile (generally 1 for horizontal or vertical movements, sqrt(2) for diagonal movement).
Each tile is then assigned an additional "heuristic" score--an approximation of the relative worth of moving to each tile. Different heuristics are used, the simplest being the straight-line distance between the centers of the given tile and end tile.
The current tile is then "closed", and the agent moves to the neighboring tile that is open, has the lowest movement score, and the lowest heuristic score.
This process is repeated until the goal node is reached, or there are no more open nodes (meaning the agent is blocked).
For diagrams illustrating these steps, refer to this good beginner's tutorialgood beginner's tutorial.
There are some improvements that can be made, mainly in improving the heuristic:
Taking into account terrain differences, roughness, steepness, etc.
It is also sometimes useful to do a "sweep" across the grid to block out areas of the map that are not efficient paths: a U shape facing the agent, for example. Without a sweep test, the agent would first enter the U, turn around, then leave and go around the edge of the U. A "real" intelligent agent would note the U shaped trap and simply avoid it. Sweeping can help simulate this.
