The arrow's fletchings work to steer it toward its intended target. This steering comes in the form of spin and it is this spin that creates a wobble in an arrow's flight.
The archer's paradox is a fascinating physics problem that occurs when an arrow goes to the right of the bowman's intended target even though it is shot from the left of the bowman. The explanation for this is quite simple and involves the arrow spinning.
Another common reason for an arrow's wobble in flight is a sloppy release. This can happen to even the most experienced archer. It's best to practice a clean release to prevent this from happening.
A sloppy release can also lead to an arrow fishtailing. This happens when the arrow has too much pressure on the nose or wraps around the bow's cheek. It can be corrected by ensuring the fletchings are properly seated and by adjusting the nocking point height.
If an arrow is wobbling up and down like the tail of a swimming porpoise it is called "porpoising." This type of wobble robs the arrow of kinetic energy. It's usually caused by an improper nocking point height and can be corrected by moving the nockset down if your arrow is fletching high or up if it is fletching low. It's also possible that the arrow spine is too weak or stiff. In this case, a video of the arrow's flight can help you determine if the spine is too weak or too stiff and take steps to correct it.