There are a lot of random scenes you can come across while playing the game. The events triggers are usually tied to the participant's Friend Point value, so the higher the FP, the more likely you will run across an event with that person. If you proceed through the event correctly then you'll further increase your FP standings with the residents involved.
The valley is divided up into 9 sections, which you can view by clicking on the World Map menu. The events you trigger can happen in any of those 9 areas, or even inside the buildings in the area. Some events also require different weather requirements or only trigger on specific days of the week.
You will be able to see a series of extra events if you plug in your copy of Friends of Mineral Town for GBA. You only need to have your (or a friend's) GBA game in your Nintendo DS console once in order to have the flag set to be able to see all of the exta events.
If you want to see the extra random events that were added into the girl version, check out this page instead.