Today a horse archer generally uses a composite recurve bow of around 50-55” (except for Japanese yabusame, which uses an asymmetric bow over 2m in length) with wooden, aluminium or carbon fibre arrows fletched with turkey feathers.
The most common way horseback archery is practised today in the UK is on a straight track, around 100 metres long with a series of targets to the left of the track, set up in either in the Korean or Hungarian style. The horse is ridden at a canter or gallop with the reins loose on the horse's neck - horses must be desensitised to the bow and arrows and the sound of shooting, as well as being trained to run at a steady speed without rein contact. Mastery of horseback archery involves the ability to nock arrows onto the string whilst moving with the horse and keeping your eye on the target, then aiming at the target (which most do in an instinctive manner, without lining up part of the bow or arrow with the target) - subconsiously adjusting for variation in distance from the target and the speed of the horse, maintaining practised archery form (despite the target being anywhere from your front, to perpendicular to your direction of movement, to behind you) and timing the shot with the horse's gait. It sounds like an awful lot to think about but it can all fall together perfectly, and when it does it the feeling really is addictive!
Less common styles of horseback archery are:
Qabaq, Turkish gourd shooting in origin, where a single blunt arrow is shot upwards at a target on an 8m high pole.
Mogu, where one or two horses and riders gallop across an open field in pursuing a mounted opponent and shooting blunt arrows dipped in paint at the 60cm diameter cloth-covered wicker ball being towed.
Yabusame, is an ancient style practised in Japan where riders shoot at very small targets with a blunt arrow shot from a long assymetric bow.
Archery along a long winding cross-country course, crossing variations in terrain and sometimes jumps (often called the Polish style).
Mamluk, where the target may be on either side of the horse, requiring forward or backward shots and the distance to the target is variable.
Jordanian, where a sword must be carried in your hand whilst executing an archery shot then it is used to pick up a target from the ground.