Popinjay is the game, arrow are called Popinjay arrows.
This some of what Wikipedia has to say about Popinjay...
The archery form of popinjay dates back to at least the fifteenth century. The annual popinjay (or papingo) tournament of the Ancient Society of Kilwinning Archers, of Ayrshire in Scotland, is believed to have been running since 1483, though the records are only reliable from 1688. The current prize of a silver arrow was introduced in 1724.[citation needed]
Popinjay archery is popular in Belgium, but is less common elsewhere. Many Belgian clubs have permanently-erected popinjay masts. Popinjay can also be shot horizontally rather than vertically, though this form is even less common.
There are no international standard rules of popinjay. The definition of rules is left to national archery organisations.
[edit] Vertical
Traditional shooting at a wooden eagle with an (bad word removed) at the "Rutenfest Ravensburg", Germany
The format and rules of popinjay given below are drawn from those defined for the United Kingdom by the Grand National Archery Society. The specific rules are given in the GNAS Rules of Shooting 2006, rules 1000 to 1006. (GNAS, 2006)
The object of popinjay is to knock artificial birds off their perches. The perches are cross-pieces on top of a 90-foot mast. The "cock" (the largest bird) is set on the top cross piece. Four smaller "hens" are set on the next crosspiece down. Two dozen or so "chicks" (the smallest birds) are set on the lower cross pieces. (GNAS, 2006 - rule 1000)
The archer stands near the base of the mast and shoots arrows upwards at the birds. (GNAS, 2006 - rule 1000) The arrows are tipped with rubber blunts rather than sharp points. The blunts are between 0.75 inches and 1 inch in diameter. (GNAS, 2006 - rule 1001)
Points are scored for each bird knocked off. Typically, the archer scores 5 points for the cock, 3 points for a hen and 1 point for a chick. (GNAS, 2006 - rule 1004)