Fortnite has a wide variety of weapons, and each weapon type is available at different rarities. From time to time, different challenges or activities will require you to use weapons of a very specific rarity to do damage or score eliminations.
Common is the most abundant weapon rarity in the game and is the easiest to find. The weapons have a grey background on their icon in your inventory and a slight grey glow when they are on the ground.
You can find common weapons from the same sources as every other weapon in the game. They can spawn on their own as ground loot all over the map or can be gotten from the various chests that will appear. Common weapons are most likely to spawn from normal wooden chests. If you are looking for a common weapon and find a rare chest, it is unlikely to have one.
Also, weapons cannot be downgraded in Fortnite. You can upgrade weapons at certain NPCs on the map, but there is no way to reduce the rarity of a weapon, so if you need one, you will need to find it.
The important thing to understand about common weapons in Fortnite is that they do the lowest damage of all the different rarities. If you are using a common pistol in a gunfight against a player with that exact same pistol but of a higher rarity, then you are automatically at a disadvantage. This can make finishing elimination challenges with them a little bit challenging.