Ludens' How to use horse archers for Medieval: Total War is still required reading for anybody interested in this type of unit. However, there are some very significant improvements in R:TW and especially in "Barbarian Invasions."
First, the "Barbarian Invasion" expansion fixes the issues that were caused — unintentionally, apparently — by the 1.2 patch, which messed up HA so that they did not fire while moving in skirmish mode. Those who don't have the expansion kit are strongly advised to use Qwerty's most excellent fix.
1. Skirmish mode works now.
2. Horse archers now fire on the move. Foot archers don't.
The effects of that are enormous. Foot archers are still one of the best counters to horse archers, but they're not nearly as good as they were in MTW, especially if the AI is handling the foot archers and a human is controlling the horse archers.
Obviously, a moving target is much harder to hit than one standing still. Therefore, the effect of firing on the move is even greater when the HA uses Cantabrian circle.
Missile battles, though, are not the HA's best use, not by a long shot.
Rain arrows upon an advancing melee infantry unit. The unit will probably keep moving and its supporting foot archers will keep moving to stay close. HA can shoot an entire phalanx to pieces before the foot archers ever get a chance to stop and fire.
Horse archers can also follow or chase units and keep firing. This shooting in the back is incredibly deadly.
3. On the strategy map, an all cavalry army with a mix of melee and HA cav can wrought havoc. They can attack several armies in one turn and are very difficult to catch. If an opposing army has any slow infantry at all, the cavalry unit can choose when and where it wants to fight.
4. The Barbarian Invasion expansion vastly increased the effectiveness of shields and armor for protecting units (in R:TW and in BI) from archer fire, but only for units that are FACING the archer fire. Units are still very vulnerable from the side or from behind. This is greatly to the advantage of HA, since they can easily and safely get to the side or behind compared to foot archers.
5. The horde HA of Barbarian Invasions now come in the same "horde" formation as peasants. This is actually a very nice feature. The horde formation has no "corners" to get caught by melee units.
[edited p.s] This is a summary of the thread's most important points to date:
1. Put HA into big square formations if they don't come in "horde" formations. This makes the unit easier to move about. HA fire in all directions and don't suffer as much from friendly fire because there is less unit overlap.
2. Beware the map's edge.
3. Try to envelop your enemy. The HA units don't have to be in contact with each other. There's no need for a "continuous front" here. Gaps between "squares" are covered by fire.
4. Move and shoot for the enemy's "weapon" side. The hand that carries the weapon has no shield.
5. Use Cantabrian circle carefully. Circling HA tire out quickly and can suffer if caught in melee. However, the tactic does reduce casualties in a pure missile fight.