Balloons occasionally float over your town with a present attached. You'll hear a wind noise when there's one nearby. If you hear the noise and can't see a balloon try pressing up on the d-pad to look up. Balloons can be shot down with a slingshot - stand under the balloon, where the shadow is, and shoot. When you open the present that was attached you may find a piece of the 8-Bit Mario Series - besides hacking or online trading balloons are the only way to obtain this series. However you may only get a regular piece of furniture.
The easiest way to find a balloon is to wait for the time to end in a 5 (X:05, X:15, X:25 etc) and simply walk up and down the eastern and western borders of your town by the cliffs. If a balloon has appeared you will either see the balloon itself or hear the wind rushing signifying it is nearby. The balloons will appear on either the 16th, 32nd or 48th tile from the top (ie on the borders of sectors as used in Perfect Town). At least one balloon appears each hour.
After you've shot down at least eight balloons you may also come across an item attached to three yellow balloons ("three-balloon" balloon). This is the Golden Slingshot. Unlike other balloon items, this item is not inside a present, but is just attached to the three balloons.