This link explains fairly well how a navigation mesh may be automatically generated from the level geometry: http://udn.epicgames.com/Three/NavigationMeshReference.html . I believe it's also a good practice for the level designer to examine it manually and make any necessary corrections for sharp corners, narrow corridors, etc.
As for the path, A* search is usually the name of the game for any kind of 2D pathfinding. With the usual Euclidean distance heuristic it always returns the shortest path, but it's possible to use an "inadmissible" heuristic to make it run faster (and produce sub-optimal paths). The arrows shown on the slides are a little baffling; I can't imagine why those four vertically stacked boxes on the right-hand side would create a jagged path, for instance. It may just be exaggerated for effect.
As for the path, A* search is usually the name of the game for any kind of 2D pathfinding. With the usual Euclidean distance heuristic it always returns the shortest path, but it's possible to use an "inadmissible" heuristic to make it run faster (and produce sub-optimal paths). The arrows shown on the slides are a little baffling; I can't imagine why those four vertically stacked boxes on the right-hand side would create a jagged path, for instance. It may just be exaggerated for effect.