At night, when the sky is clear, you might see a shooting star streak across the sky, accompanied by a twinkling sound. When this happens, wish on the star and you will get a gift in the mail the next day from Wishy the Star. Wishy will give you items that you don't often see in Tom Nook's store.
Shooting stars might appear in the sky between 7PM and 9PM. The sky needs to be clear, with no clouds at all, for the shooting star to appear. However, shooting stars are completely random, and sometimes you might not see a shooting star for a very long time. But sometimes, there are nights when there are large numbers of shooting stars all throughout the night. This is often referred to as a meteor shower.
When you hear a shooting star and see it streaking across the sky, you can press A, or tap your person, and the person will clasp his or her hands and bow down, making a wish. If your person does not bow, then you did not wish successfully. You need to make the wish while the star is still in the sky, before it goes off of the edge of the screen.
When you wish on a star, Wishy the Star will fulfill your wish by sending you a letter that will arrive in your mailbox in the morning. Attached to the letter there will be a gift that is a Spotlight Item: a rare item that might sometimes (but rarely) appear in Tom Nook's store. There are some very valuable Spotlight Items, so getting them for free from Wishy is a great deal.
No; you can only get one gift from Wishy per day. Even if you wish a bunch of times a night, like during a meteor shower, you will only receive one gift the next day.
Yes. Friends who visit your town via Wi-Fi or the Nintendo WFC can wish on a shooting star while they are in your town, and the gift from Wishy will appear in their mailboxes in their own towns.
The shooting stars are random, as far as I know. There might be some secret schedules programmed into the game, but they probably are still pretty random. It is possible that the designers of the game made it more likely for shooting stars to appear during certain times of the year, but it is also possible that it is completely random. Nobody really knows!