Archery is one of the oldest Olympic sports, but the physics involved are remarkably complex. Even looking only at the flight of the arrow, the problem is hardly simple. The heavy point of the arrow makes it front-heavy, and the fletches on the back of the arrow provide additional surface area on which air can act. This means that the center of mass of the arrow–where gravity acts–is further forward than the center of pressure–where aerodynamic forces act. This results in the aerodynamic forces helping to stabilize the flight of the arrow. To see why this is important, try throwing a dart fletching first!
When an arrow is fired from a bow, as in the high speed video above, the sudden impetus of force from the bowstring causes the arrow to flex and vibrate as it is fired. The aerodynamic forces generated by the fletches straighten the arrow’s flight, helping it reach the intended target accurately. Some fletching is designed to make the arrow spin; this can further improve accuracy but comes at the cost of speed since some of the arrow’s initial kinetic energy must be converted to rotation. For more, check out Archery Report, which features some great articles on the physics of archery and even has CFD comparing arrow tips. Mark Leach also has some great information on tuning a bow, which, if done properly, allows one to accurately shoot unfletched arrows.